Presented to the
by the
ONTARIO LEGISLATIVE
LIBRARY
1980
NAVIGATION
THE MACMILLAN COMPANY
NEW YORK • BOSTON • CHICAGO • DALLAS
ATLANTA • SAN FRANCISCO
MACMILLAN & CO., LIMITED
LONDON • BOMBAY • CALCUTTA
MELBOURNE
THE MACMILLAN CO. OF CANADA, LTD.
TORONTO
NAVIGATION
BY
HAROLD JACOBY
RUTHERFURD PROFESSOR OF ASTRONOMY
IN COLUMBIA UNIVERSITY
SECOND EDITION
WITH A CHAPTER ON COMPASS ADJUSTING AND A
COLLECTION OF MISCELLANEOUS EXAMPLES
ELECTRONIC VERSION
AVAILABLE
NO.
THE MACMILLAN COMPANY
-7,-isf " vmmm
p. — -^
"2 /
VK
\ <£, V N
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COPYKIGHT, 1917 AND 1918,
BY THE MACMILLAN COMPANY.
Set up and electrotyped. Published October, 1917.
Second edition, with new matter, February, 1918.
Norfoooti
J. S. Gushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
Co
MACLEAR JACOBY
QUARTERMASTER,* THIRD CLASS, U. 8. N.
BNLI8TED FOR THE PERIOD OF THE WAR
THIS VOLUME IS OFFERED AS
A MARK OF RESPECT
BY HIS FATHER
* COMMISSIONED ENSIGN, U. S. N. R. F., SEPTEMBER, 1917
THE present volume was undertaken with certain very
definite aims. In the first place, it is intended to be com-
plete in itself, so that it should be possible to navigate a ship
in any ocean not very near the north or south pole without
other books or tabular works, excepting only the nautical
almanac for the year in which the voyage is made. To attain
this end without unduly extending the size of the volume,
certain essential nautical tables have been abridged; but
all are given in sufficiently extended form to permit of actual
navigation with their aid; and they are especially suitable
for beginners, who can here attain the necessary knowledge
with less effort than would be necessary with more bulky
volumes. In cases where very extended tables are conven-
ient, they are mentioned in the text.
In the second place, the author has not assumed that the
reader possesses formal mathematical and astronomical
knowledge, or desires to possess such knowledge. When-
ever methods of navigation require for their demonstration
an understanding of spherical trigonometry, or some other
branch of formal mathematical science, such demonstrations
have been replaced with incomplete or "outline " demonstra-
tions designed for the non-mathematical reader. Practical
methods are fully explained ; and an attempt has always
been made so to word the explanations that the reader,
even the beginner, will understand his problem, and will
know what he is doing, and why he does it.
The requirements of those who may study without a
teacher have received constant and special attention. To
meet these requirements the whole subject is presented in
vii
viii PREFACE
a somewhat informal manner; such topics as the use of
logarithms, or the principles on which all mathematical
tables are constructed — these less attractive parts of the
subject are not presented in a special chapter, but are de-
scribed in a sort of digression, when needed in the discussion
of an actual navigational problem.
Finally, to further simplify and condense his material,
the author has made no attempt to include every method
that can possibly be used to navigate a ship, or that ever has
been used to navigate a ship ; his purpose has been rather
to limit the volume to the methods at present thought best
by the most reliable modern authorities.
Other books on navigation have been used freely, espe-
cially in the preparation of the tables. Among these, that
admirable encyclopedia of navigation, known as "Bowditch,"
published by the Hydrographic Office, United States Navy,
and Kelvin's "Tables for Sumner's Method at Sea" have
been found of the greatest help.
Miss Dorothy W. Block, Instructor of Astronomy in
Hunter College, New York, has helped with great energy
in the preparation of the tables and the correction of the
text. It is hoped that suph errors as may now remain in
the book are few in number.
H. J.
COLUMBIA UNIVERSITY,
August, 1917.
PREFATORY NOTE TO THE SECOND EDITION
To meet the wishes of certain young navigators, this edition
has an added chapter on the adjustment of correctors in a
compensated compass binnacle, and also a collection of new
problems and examples.
H. J.
February, 1918.
TABLE OF CONTENTS
HAPTEK PAGH
I. THE FUNDAMENTAL PROBLEM OF NAVIGATION . . 1
The problem stated. Reasons for the existence of the
problem. Definition of "ship's position." Longitude
meridians and latitude parallels. Greenwich the initial
meridian. Position determined by observation; on the
coast and at sea. Dead reckoning. Sextant observa-
tions. Chronometer.
II. DEAD RECKONING WITHOUT LOGARITHMS . . > .,.: .7
The two problems. Designation of .ship's course.
Latitude difference and departure. The traverse table.
Use and construction of tables in general. Arguments
and tabular numbers. Relation between departure and
longitude difference. Middle latitude.
III. DEAD RECKONING WITH LOGARITHMS ... 23
Explanation of number logarithms and their use.
Multiplication and division. Trigonometric logarithms.
Solution of the two problems. Middle latitude sailing.
Mercator sailing. Meridional parts. Great circle sail-
ing. The rhumb line. Composite sailing. Parallel
sailing. Traverse sailing.
IV. THE COMPASS 40
The card, how divided. Degrees and points. Boxing
the compass. Lubber line. True course and compass
course. Error, variation and deviation. Swinging ship.
Azimuth circle and pelorus. The compass formulas.
The two deviation tables. Comparative table of points
and degrees.
V. COASTWISE NAVIGATION ...... 53
The "fix." Bow bearings. Doubling the bearing on
the bow. Bow and beam bearings. Distance a-beam.
Cross bearings. The danger angle. Danger bearing.
Soundings.
ix
X TABLE OF CONTENTS
CHAPTER PAGE
VI. THE SEXTANT . 61
Description of the instrument and its use. The vernier.
Index error. Three adjustments. The artificial horizon.
Correcting the altitude. Dip. Refraction. Parallax.
VII. THE NAUTICAL ALMANAC ...... 75
Specimen pages of it. Greenwich mean time. Decli-
nation. Equation of time. Astronomic and civil day.
Apparent solar time. Chronometers and the rate card.
Right ascension. Solar and sidereal time.
VIII. OLDER NAVIGATION METHODS 86
The noon-sight for latitude. Tropic observations and
the midnight sun in high latitudes. Preparing for the
observation. Setting the cabin clock. Star observa-
tion. Ex-meridian observation. The time-sight for
longitude. Set of current. Star time-sight. Condensed
forms of calculation.
IX. NEWER NAVIGATION METHODS ..... 108
Errors produced by dead reckoning. Captain Sumner,
and the Sumner line. Bearing of the line. The Sumner
point. Azimuth tables. Condensed form of calcula-
tion. Star observations. Comparison of Sumner navi-
gation with time-sight navigation. The Kelvin table.
Condensed forms of sun and star observations. Inter-
section of two Sumner lines obtained with a special table.
Motion of ship between observations.
X. A NAVIGATOR'S DAY AT SEA ..... 141
Voyage planned from New York to Colon. Departure
at Sandy Hook lightship. The course to Watlings Island.
The variation and deviation applied. Azimuth of the sun
observed at sunrise. Bow and beam bearings of Barnegat
Light. The patent log and the log book. New course
from Barnegat. Morning sight worked as a Sumner line.
„ Another Azimuth observation. Weather thickens at
11 : 30. Ex-meridian sight at 11 : 42, worked as a Sumner
line. Afternoon sight worked as a Sumner line. Posi-
tion of ship fixed from intersection of the two lines. East-
erly current estimated. Compass error again tested.
The course set for the night.
TABLES . . . . . . . . . .153
APPENDIX 1. Compass Adjusting . . . . . 323
APPENDIX 2. Miscellaneous Examples ..... 335
LIST OF ABBREVIATIONS
USED IN THE PRESENT VOLUME
Alt. for altitude ;
App. for apparent ;
Arg. diff . for argument difference ;
Cf . for compare ;
Chron. for chronometer ;
Comp'd for computed ;
Cos for cosine ;
Cot for cotangent ;
Csc for cosecant ;
C. — W. for chronometer minus watch ;
Dec. for declination ;
Dep. for departure ;
Dist. for distance ;
D. R. for dead reckoning;
Eq. for equation of time ;
G. A. T. for Greenwich apparent time;
G. M. T. for Greenwich mean time ;
Hav. for haver sine ;
H. D. for hourly difference ;
Int. diff. for interpolation difference ;
Lat. for latitude ;
Lat. diff. for latitude difference ;
Log for logarithm ;
Long. for longitude ;
Long. diff. for longitude difference ;
Mer. lat. diff. for meridional latitude difference ;
Obs'd for observed ;
p for polar distance ;
R. A. for right ascension ;
s for hah3 sum ;
Sec for secant ;
Sin for sine ;
T for ship's apparent solar time (or star's hour-angle) ;
Tab. diff. for tabular difference ;
Tan for tangent.
xi
NAVIGATION
CHAPTER I
THE FUNDAMENTAL PROBLEM OF NAVIGATION
To find one's way in a ship across the trackless ocean is
our problem. Most people would like to know how it is
solved ; nor is the solution very difficult to understand when
set forth in simple language and without too great wealth of
technical detail. We hope the reader will find this to be
the case after a study of the following pages.
Our fundamental problem can be more fully stated quite
easily. It consists in the determination of a ship's location
on the earth's surface at any given moment. If this loca-
tion can be determined, it becomes a comparatively easy
matter to ascertain the direction (north, south, northeast,
southeast, etc.) in which the ship must be steered in order
to reach her port of destination. For the location of the
port of destination on the earth's surface is of course also
known : and if we know where the ship and her destined port
both are, we can easily find the right course for the helmsman.
With the fundamental problem stated in this way, it
would almost seem as if there were really no such problem
in existence. For when the ship begins her voyage, she is
necessarily in a known port. Knowing also the port to
which she is to go, we should be able to determine her proper
course from the one known port to the other. This course
being then steered, no further navigational proceedings would
be required. But this reasoning is incorrect, because a ship
B 1
2 NAVIGATION
does not actually advance across the ocean in exactly the
direction in which she is steered. Ocean currents deflect
her ; and the action of a strong wind blowing against one of
her sides will have a similar effect. Currents and winds
cannot be predicted with accuracy : and so it becomes
necessary to re-determine the ship's position frequently at
sea. This should be done at least once daily if possible;
and when it has been done, the mariner can take a new
"departure," as he calls it, and lay a new course for his
intended port. Thus the effect of ocean currents, etc., can
be eliminated, and the voyage made as safely as if they did
not exist.
Now this determination of the ship's position at sea,
and when out of sight of land, is strictly an astronomical
problem. It can be solved by means of astronomical ob-
servations, and in no other way. But before giving an out-
line of how this is done, let us first see what is meant by
the words "ship's position at sea." How can we describe
a ship's position so that one mariner could tell another
where she is located, and thus enable the second mariner to
find her?
To thus indicate the point on the earth's surface occupied
by the ship has a certain similarity with giving the address of
a house in a city. Such a city address always consists of
two separate statements; as, for instance, the name of a
street and the number of the house. An address cannot
be given completely unless two different facts are stated.
They need not necessarily be a street name and a street
number : we can equally well designate such an address by
stating that the house is at the corner of a certain street and
a certain avenue. But here also the address is made up of
two separate facts.
This form of stating an address as the intersection of a
certain street and avenue is the form having the closest
^resemblance to the method of the navigator. If the city
avenues are supposed to run north and south, and the streets
THE FUNDAMENTAL PROBLEM OF NAVIGATION 3
east and west, as they do in New York (approximately), the
analogy with navigation will be almost perfect.
For the navigator imagines the earth covered with a net-
work consisting of "avenues, " running north and south, and
"streets," running east and west. He calls the "avenues"
meridians of longitude, and the " streets " parallels of latitude.
Then he designates the position of a ship on the ocean by
stating that it is at the intersection of a certain meridian
of longitude and parallel of latitude. There are 360 such
meridians of longitude : each begins at the terrestrial equator,
and runs north and south from there to the north and south
poles of the earth. Of the latitude parallels there are ISO.1
They all run east and west, parallel to the terrestrial equator ;
90 are between the equator and the north pole, and the other
90 between the equator and the south pole.
One of the longitude meridians (that passing through
Greenwich, England) is chosen arbitrarily as the starting
point for counting longitude meridians. To this initial
meridian is assigned the number 0, and the other meridians
are numbered successively 1, 2, 3, etc. So numbered,
the meridians are called "degrees" of longitude; the third
one, for instance, being written 3°. The meridians may be
counted either eastward or westward from Greenwich, a
ship on the 20th meridian west of Greenwich, for instance,
being in longitude 20° west.
The latitude parallels are similarly counted north and
south from the equator ; and if the above ship were on the
40th latitude parallel north of the equator, her complete
"address," or position at sea, would be long. 20° W. ; lat.
40° N.
Of course a ship would only rarely be located exactly at
the intersection of a meridian and parallel. Therefore, the
space between any two successive meridians and between
any two successive parallels is subdivided into 60 parts,
called minutes of arc. Thus the above ship, if halfway
1 Including the equator twice, but excluding the two poles.
4 NAVIGATION
between a pair of meridians and also halfway between a
pair of parallels, might be in longitude 20° 30' west, and
in latitude 40° 30' north. This would be written long.
20° 30' W. ; lat. 40° 30' N.
Each minute of longitude and latitude is further sub-
divided, when extreme accuracy is required, into 60 seconds ;
so that if the ship were a little to the north and a little to
the west of the above position, she might, for instance, be
in long. 20° 30' 26" W. ; lat. 40° 30' 10" N.
These meridians and parallels, or longitude and latitude
lines, appear on many maps and charts as straight lines,
or at least as lines only slightly curved. But being all lines
imagined drawn on the earth, which is almost an exact
sphere or round ball, they must really all be circles. Thus,
the terrestrial equator is really a big circle, girdling the
earth, and divided into 360 equal parts, or degrees. At
each of the division points a meridian starts northward
toward the pole. This meridian is also a big circle
perpendicular to the equator. The distance along the
meridian from the equator to the pole is divided into 90
equal parts or degrees, and the whole distance from equator
to pole is one quarter of a complete circumference of the
earth. The 90 degrees, from equator to pole, thus repre-
senting one quarter of a circumference of the earth, a com-
plete circumference contains 4 X 90, or 360 degrees, the
same as the equator. So the degrees measured along the
meridians are equal to the degrees measured along the
equator. The former are degrees of latitude, the latter
degrees of longitude; and degrees of latitude are equal to
degrees of longitude, when the latter are measured along
the equator. The length of each degree is then 60 nautical
miles.
Having thus indicated what is meant by a ship's position
in latitude and longitude, we shall next describe in outline
how such a position may be determined by observation.
Tf the ship is within sight of a coast-line, there will probably
THE FUNDAMENTAL PROBLEM OF NAVIGATION 5
be some lighthouse, or other "aid to navigation," in view,
from which the navigator can ascertain where he is. Methods
for doing this are described later (p. 53). But when -the
ship is really at sea, with no land in sight, real deep-sea
methods must be employed.
These methods, when the weather is clear, always include
an observation of the sun or some other heavenly body.
When the weather does not permit such observations, the
mariner can still find his position approximately by means
of "dead reckoning" (abbreviated, D. R.). This process
will be described in detail in the next chapter; but we can
already state that it consists in a calculation based on his
astronomic observation of latest date. Knowing where the
ship was the last time he observed the sun, and -also know-
ing both the direction in which he has steered and the
(approximate) speed of the ship, the navigator can calculate
(also approximately) the location of the point he has reached.
Even when astronomical observations are made, the
D. R. calculation is always carried out, because the navi-
gator is always anxious to know how nearly correct his
D. R. result would have been, if the day had been cloudy.
Furthermore, this result also acts as a check on the astronomi-
cal work, and tends to increase the navigator's confidence
in the correctness of his final result as to the ship's location.
The manner in which the ship's position is found from
astronomic observations will of course be explained in detail
later. It is all done with an instrument called a sextant.
This is merely a contrivance with which the navigator can
measure how high the sun (or other heavenly body) is in the
sky at any moment. The sun is highest in the sky daily
at noon, but it is not equally high on different days in the
year. Nor is it equally high on the same date in different
latitudes. Thus, by measuring with the sextant how high
it is on any particular date at noon, as seen from the ship,
the navigator learns the terrestrial latitude in which the
ship is located.
6 NAVIGATION
Similar sextant observations made at other suitable times
during the day, when combined with exact readings taken
from an accurate chronometer such as every ocean-going
ship carries, will similarly make the ship's longitude known,
All this will of course be explained in full detail in later
chapters.
CHAPTER II
DEAD RECKONING WITHOUT LOGARITHMS
As we have seen (p. 5), this is a process by means of
which the mariner can calculate a ship's position in latitude
r e
\
0' 5
/Vest Lc
9° 5
jngitud*
8° 5
\
r 5
6° 5
sr
46°
45°
44'
NJ
•s/«r P
W/-/
/o /
43°
1
/
42
41'
40'
Fio. 1. — Dead Reckoning. (Diagram not drawn to scale.)
7
8 NAVIGATION
and longitude, without special astronomic observations of
any kind. In the accompanying Fig. 1, which represents a
portion of a chart of the North Atlantic, a ship's position
at noon is shown at the point Y. This point we will call
the ship's "initial position," in discussing our present prob-
lem. We will suppose that it was correctly obtained by as-
tronomic observations, and that these showed the ship at Y
to be in lat. 42° 11' N. and long. 59° 28' W. from Green-
wich. Sometime in the afternoon, having traveled a dis-
tance estimated from the known speed of the ship as 63 miles,
and having "made good" this distance in the direction YP,
the ship arrives at P. This point P we will call the ship's
"final position" ; and our problem now is to find its latitude
and longitude.
This problem may be called the first fundamental dead-
reckoning problem. The second and remaining fundamental
problem is the converse of the first, and may be stated as
follows : having given the latitude and longitude of the initial
point Y, as occupied by the ship, and also the latitude and
longitude of the final point P, it is required to find the dis-
tance from Y to P in miles, and also the direction of the line
YP.1
To understand these two problems properly it is next
necessary to explain how we may define the words "direc-
tion YP." This is done by referring the line YP to the
direction of the arrow shown in the figure. This arrow
is parallel to the longitude meridians on the chart, and
therefore points due north. The angle between the arrow
YN and the line YP is marked in the figure, and is called
the "ship's course." This angle is really the difference in
direction of the two lines YN and YP. The point Y is called
the "vertex" of the angle, and all angles are designated
1 We think it advisable to place these two important converse
problems together, and to call them both pro.blems^of. dead reckon-
ing, though many writers on navigation confine the phrase " dead
reckoning" to the first fundamental problem alone.
DEAD RECKONING WITHOUT LOGARITHMS 9
FIG. 2. — Dead
Reckoning.
by three letters, the letter belonging to the vertex being
placed between the other two; in this case the angle is
called either NYP or PYN.
Now let us draw a line PQ (fig. 2), from P to NY, and
perpendicular to NY. Then the motion of the ship from
Y to P will have carried her north of the N,,
point Y by a distance equal to YQ, and east
of the point Y by a distance equal to QP. Q
This is not strictly true, unless the earth's
surface, throughout the small area involved
in the present problem, can be regarded
as a flat surface. Such a flat surface is
called in geometry a "plane" surface; and
these calculations therefore belong to that
part of navigation which is called "plane sailing." Plane-
sailing calculations are easy calculations, and they are
generally sufficiently accurate for the purposes of the
navigator.
The ship's course, being thus an angle, must be designated
by means of a unit of measure
suitable for measuring angles.
For this purpose the degrees and
minutes already used for longi-
tude and latitude (p. 3) are
usually employed. Fig. 3 shows
that a latitude, for instance, is
really an angle, and must there-
fore also be measured in de-
grees. P is the earth's pole, PQ
a meridian, and the latitude of
the observer at 0 is the angle
OCQ, here about 40°.
So it is clear that the ship's course NYP (figs. 1 and 2)
will be measured in degrees. Minutes are not really needed
in measuring courses, as they are in measuring latitudes;
the nearest whole degree is always accurate enough, because
FIG. 3. — Latitude Angle.
10 NAVIGATION
it is never possible to steer a ship on her proper course with
absolute exactness. In fact, many mariners use a still less
precise method of measuring courses by means of "the points
of the compass." (See p. 40.)
Resuming our two fundamental problems (p. 8), let us
now begin with the first one, and proceed to find the lati-
tude and longitude of the point P (figs. 1 and 2). To solve
this problem, we must not only know the distance YP
(63 miles), as traveled by the ship, but also the number of
degrees in the course angle NYP. Let us suppose this course
angle happens also to be 40°. The problem
then appears as shown in Fig. 4. We now
know the distance YP and the angle QYP.
Evidently the next step is to find the distances
QFand QP. QY, in our present problem, is
called a "latitude difference" and QP is called
a "departure."
FIG. 4. -Dead To find the "latitude difference" and
"departure" from the course angle and dis-
tance we may either use that branch of mathematics called
plane trigonometry, or we may find them from a special
navigation table, called a "traverse table." Our Table 1
(beginning p. 154) is such a table.
Before l beginning its use it will be well for the reader to
note in general that all mathematical tables consist of two
sets of numbers. The first set of numbers are called " argu-
ments" of the table, and the second set are called "tabular
numbers." The main object of the table is to furnish us
with the proper tabular number when we know the proper
argument.
The ordinary multiplication table is a good example of a
mathematical table. It is usually .written as follows and
1 The beginner may find it advisable, on a first reading of the
book, to omit this explanation of mathematical tables, returning
later when he finds a reference to it in the text. The dead reckoning
problem under discussion is resumed on p. 13.
DEAD RECKONING WITHOUT LOGARITHMS 11
it affords a good opportunity of studying the principles
underlying all mathematical tables in a case so simple as
to offer no difficulty.
MULTIPLICATION TABLE
(to illustrate " argument" and " tabular number")
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
2
4
6
8
10
12
14
16
18
20
22
24
3
6
9
12
15
18
21
24
27
30
33
36
4
8
12
16
20
24
28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
55
60
6
12
18
24
30
36
42
48*
54
60
66
72
7
14
21
28
35
42
49
56
63
70
77
84
8
16
24
32
40
48
56
64
72
80
88
96
9
18
27
36
45
54
63
72
81
90
99
108
10
20
30
40
50
60
70
80
90
100
110
120
11
22
33
44
55
66
77
88
99
110
121
132
12
24
36
48
60
72
84
96
108
120
132
144
In this table the arguments are printed in heavy type and
are contained in the left-hand column and the topmost
horizontal line. In using the table, these arguments are
given in pairs, being always the pair of numbers to be mul-
tiplied. In fact, in the case of most tables, the arguments
are thus given in pairs, though there are some tables with
but a single argument. In the present case one number
from the pair of arguments will be found in the left-hand
column, the other in the top horizontal line. Thus, if we wish
to multiply 6 and 8, these two numbers constitute the pair
of arguments. We find the right line (belonging to 6) and
column (belonging to 8), and the tabular number 48 (marked
with a *) occurs at the intersection of the 6-line and the 8-
column. If the pair of arguments are taken in the order
8x6 instead of 6 X 8, we should use the 8-line and the
6-column, again finding the required product (48) as the
tabular number at the intersection.
12 NAVIGATION
Sometimes the given arguments cannot be found di-
rectly in the table. Thus we might wish to multiply
6| (written 6.5) by 8. Evidently the proper tabular
number would be halfway between the 6x8 tabular
number (48) and the 7x8 tabular number (56). The
correct answer would therefore be 52. This process, by
which the tabular number 52 is obtained, is called "in-
terpolation." The example 6| X 8 is an extremely simple
one. When less easy ones occur, the interpolation is best
made as follows : we ascertain by subtraction how much
the tabular number increases while the argument changes
from 6 to 7. This increase is here 8, because the tabular
number changes from 48 to 56 in the 8-column, while the
argument in the left-hand column changes from 6 to 7.
This increase of 8 in the tabular number is called a "tabular
difference." We now compare the given argument (6.5)
with the nearest argument (6) occurring in the left-hand
column of arguments, and find an "argument difference"
of 0.5 (being 6.5 minus 6). Since this "argument dif-
ference" is 0.5, we must evidently take 0.5 X 8 (8 being the
tabular difference), and increase the tabular number 48 by
0.5 X 8, or 4. This again brings us to 52. Similar exam-
ples are :
(1) 5.3 X 4 = 21.2 ; (2) 7.7 X 8 = 61.6.
In example (1) the tabular numbers are 20 and 24 ; the
tabular difference is 4. 0.3 X 4 = 1.2; 20 + 1.2 = 21.2, the
answer. Both examples may be verified, of course, by ordi-
nary multiplication.
When both given arguments contain fractions, as, for
instance, 5.3 X 8.4, the resulting "double interpolation"
is so complicated as to be of little practical use to the navi-
gator.
To make this general explanation of mathematical tables
complete, it remains to show how they can be used in an
inverse manner ; i.e. to find the argument from the tabular
DEAD RECKONING WITHOUT LOGARITHMS 13
number. Thus, if we were told that the tabular number is
48, and one argument 8, an inspection of the table would
at once show that the other argument must be 6. In this
way the table might be used for division as well as multi-
plication ; and interpolation would evidently also be possible.
Many " mathematical tables must frequently be thus used
in an inverse manner.
Having thus explained the peculiarities of mathematical
tables, we return to our dead-reckoning problem and its
solution by means of the traverse table (p. 154).
Referring to that table we find a column (p. 167),
headed 40°, the course angle of our present problem. On
the left-hand side of the page we find the given distance, 63.
Then, opposite the distance 63, and under 40°, we find the
latitude difference (abbreviated, "Lat.") and the departure
(abbreviated, "Dep.") to be:
lat. = 48.3, dep. = 40.5.
The following are additional examples for practice :
Given : dist., 84, course 26° ; Ans., lat. = 75.5, dep. = 36.8.
Given : dist., 28, course 11° ; Ans., lat. = 27.5, dep. = 5.3.
When the course is between 1° and 45° the course angle
will be found in Table 1 at the head of the column : but when
the course is between 45° and 90°, it appears at the foot of
the column. In the latter case, the tabular lat. and dep.
are to be taken from the columns having "Lat." and "Dep."
at the foot instead of the top of the column. Examples
follow :
Given : dist., 63, course 50° ; Ans., lat. = 40.5, dep. = 48.3.
Given : dist., 84, course 64° ; Ans., lat. = 36.8, dep. = 75.5.
Given : dist., 28, course 52° ; Ans., lat. = 17.2, dep. = 22.1,
In addition to the course angles from 1° to 90°, three ad-
ditional angles are given in parentheses at the top and foot
of each column. Thus, with the course angle 30° appear
also 150°, 210°, 330°. This simply means that the latitudes
14
NAVIGATION
and departures are the same for these four
course angles. The accompanying Fig. 5 shows,
for instance, that the departures QP and Q'P'
are equal for 30° and- 150° courses if the two
distances YP and YP' are alike.
It will be noticed also that our traverse table
always gives distances from 1 to 50 on a left-
hand page, and from 50 to 100 on a right-hand
page. When distances larger than 100 occur,
it is necessary to use the 100, 200, etc., given on
the lower part of each page. If, for instance,
we require the latitude and departure for a
distance 363 miles, course 40°, we turn again to
the 40° column, and find (near the bottom of
30° and 150°. the page) :
For 300 miles, lat. = 229.8, dep. = 192.8
and (in the usual way) for 63 miles, lat. = 48.3, dep. = 40.5
Sums, 363 =278.1 233.3
Consequently, for dist. 363, course 40°, lat. =278.1, dep. =233.3.
Other examples are :
Course 25°, dist., 452 ; lat. = 409.6, dep. = 191.0.
Course 68, dist., 521 ; lat. = 195.2, dep. = 483.1.
Course 226, dist., 384 ; lat. = 266.8, dep. = 276.2.
When the given distances or course angles, which are
really the "pairs of arguments" (p. 11) of the traverse table,
contain fractions, interpolation can be used ; but such close
accuracy is seldom, if ever, required in navigation.
More extended traverse tables will be found in Bowditch's
"American Practical Navigator," published by the Navy
Department, Washington. They are also printed separately
in Bowditch's "Useful Tables." Both volumes can be
purchased at any "navigation shop " where instruments
and books suitable for navigators are sold.
To complete this explanation of our traverse table, it is
still necessary to mention that it also provides, with suf-
ficiently close approximation, for the method of measuring
DEAD RECKONING WITHOUT LOGARITHMS 15
course angles in "points of the compass" (pp. 10, 41). This
method is not now in use in the United States Navy, but it
is still largely employed in merchant vessels. It is sufficient
to state here that a course of 3 points, for instance, is very
nearly equal to a course of 34°, and the traverse table column
for 34° may properly be used for a 3-point course. Similarly,
31° may be used for 2f points, and the mariner desiring to use
points can always find from the traverse table itself just
what column to use. A special traverse table for points may
also be found in Bowditch's Tables, already mentioned.
We have now shown how to find latitude difference and
departure by means of the traverse table. But our problem
is not yet completely solved. Our ship (p. 8) started from
the point Y in lat. 42° 11' N. ; long. 59° 28' W. She traveled
63 miles on a 40° course, and the traverse table showed that
she thus made good a latitude difference of 48.3 miles and a
departure of 40.5 miles. It now remains to ascertain how
much the ship changed her latitude in degrees and minutes
from 42° 11' N. and her longitude in degrees and minutes
from 59° 28' W. When we have found these last changes,
we can learn the latitude and longitude of the point P,
which we are required to find.
To get the latitude change in degrees and minutes from
the latitude difference in miles offers no difficulty. If the
miles used are nautical miles (and in navigation they always
are nautical miles), each mile of latitude difference corre-
sponds to 1' of angular measure (p. 9), and 60 miles corre-
spond to 1°. Thus our ship must have changed her latitude
48'.3, corresponding to a latitude difference of 48.3 miles.
Her initial latitude having been 42° 11' N., her final latitude
at P will be 42° 11' + 48' (if we omit the odd .3) or 42° 59' N.
The relation between departure and difference of longitude
is not quite so simple. Our ship's departure of 40.5 miles
might correspond to far more than 40.5 minutes of longitude.
In fact, in very high latitudes near the north pole, the longi-
tude meridians converge so closely that a person traveling
16 NAVIGATION
a few miles might change his longitude very greatly. At the
pole itself a man might change his longitude 180° by simply
stepping across the pole. So it follows that the longitude
difference in minutes is greater than the departure in miles
(however, cf . p. 4) . The difference between the two increases
rapidly as we approach high latitudes though it is nil at
the equator; in Table 2 (beginning p. 168) we give this
excess of longitude difference over departure for all latitudes
under 60°, and for all longitude differences up to 100. When
the longitude differences are greater than 100, it is necessary
to use the numbers given for 100, 200, 300, etc., near the
bottom of each page in the table, and to sum tabular num-
bers, precisely as we did with the traverse table.
It will be noticed that Table 2 gives "tabular numbers"
for each degree of latitude in a separate column, and that
these various latitudes are called "middle latitudes." Thus
the middle latitude and the longitude difference are the pair
of arguments (p. 11) for Table 2, and, as we shall see pres-
ently, the use of the middle latitude avoids any uncertainty
in choosing the correct column for use. In our present
problem we have at our disposal (p. 15) two different lat-
itudes : the initial latitude at the point F, 42° 11' N., and
the final latitude at the point P, 42° 59' N. In this case, the
two latitudes are so nearly equal that we might use either
of them as an argument in Table 2 without material inaccu-
racy. In fact, in using Table 2 it is unnecessary to consider
minutes of latitude, the nearest degree being sufficient.
But often the two latitudes available at this stage of the
problem differ by many degrees. In such cases mariners
always use the average of the two latitudes, and call it the
"middle latitude." In the present case, the middle latitude
would be found thus :
Initial latitude = 42° 11'
Final latitude = 42° 59'
Sum = 85° 10'
\ sum = middle latitude = 42° 35'
DEAD RECKONING WITHOUT LOGARITHMS 17
The nearest even degree to 42° 35' is 43°, and the prob-
lem would therefore be worked with the 43° column of middle
latitude in Table 2.
Before completing our problem it is necessary to point
out that while Table 2 is intended primarily for changing
longitude differences in minutes into departures in miles, it
can also be used (as stated at the foot of each page) for the
inverse transformation of departures into longitude dif-
ferences ; and this is the transformation we must make in
our present problem. It is merely necessary to use the
departure (40.5) in the left-hand column, at the head of
which are the words "Long. Diff. or Dep.," indicating that
either of these two may be used as the argument in that
column. Then, in the 43° column of middle latitude, we
find (using interpolation) the tabular number 10.8.
This means that a longitude difference of 40'. 5 corre-
sponds to a departure of 40.5 — 10.8 miles, or 29.7 miles.
But when the table, as in the present case, is used for the
inverse transformation, the tabular number 10.8 must,
before use, be multiplied by the factor given at the bottom
of the column. For the middle latitude 43° this factor is
1.37; and so the right tabular number becomes, in the
present case :
10.8 X 1.37 = 14.8;
and as the longitude difference is always greater than the
departure, it follows that the departure of 40.5 miles gives
a longitude difference of :
40.5 + 14.8 = 55'.3 = 0° 55',
if we omit the odd tenths.
The initial longitude of the ship at the point Y was
59° 28' W. As her 40° course has carried her nearer to Green-
wich, it follows that her final longitude at the point P is :
59° 28' W. - 0° 55' = 58° 33' W.
We shall now discuss the following similar problem :
A ship takes her departure from a point about one mile
c
18 NAVIGATION
east of Navesink Highlands Light, New Jersey, in the initial
lat. 40° 24' N., initial long. 73° 58' W., and travels 1377
miles on a course of 166°. What final latitude and longitude
does she attain ?
Entering the traverse table in the column headed 166°,
which is the same as the 14° column, we find :
For dist. 900, lat., 873.2, dep., 217.7
For dist. 400, lat., 388.1, dep., 96.7
For dist. 77, lat., 74.7, dep., 18.6
Sums, 1377, 1336.0, 333.0
To make the large given distance (1377 miles) come within
the range of Table 1, it has been necessary to enter the 166°
column three times, with the arguments 900, 400, and 77,
and then to sum the corresponding tabular numbers.
The latitude difference, 1336 miles, is equivalent to 1336',
or 22° 16', counting, as usual, 60' to 1°. Then, since the
direction of her course (166°) carried the ship to the south
of her initial position (cf. Fig. 5, p. 14, and p. 19), we have :
Initial lat., 40° 24' N.
Lat. diff., 22° 16' N.
Final lat., 18° 8' N.
Middle lat., 29° 16' N.
Now turning to Table 2, hi the proper column for middle
latitude 29° :
For dep. 300 tabular number is 37.6
For dep. 33 tabular number is 4.1
Sums 333 41.7
As in the former example, this 41.7 must be multiplied
by the factor at the bottom of the column. This factor is
1.14. Multiplying, we have: 41.7 X 1.14 = 47.5. Conse-
quently, long. diff. = 333 + 47.5 = 380'.5 = 6° 20'.5. Since
the direction of her course (166°) carried the ship eastward,
and therefore nearer to Greenwich, it follows that her final
longitude is 73° 58' W. - 6° 20', or 67° 38' W. The final
position is therefore : lat. 18° 8' N. ; long. 67° 38' W.
DEAD RECKONING WITHOUT LOGARITHMS 19
' The point indicated by this final latitude and longitude
is just off the entrance to the Mona Passage, between Haiti
and Porto Rico ; the given course and distance would there-
fore be correct for a voyage from New York to Mona Passage
Additional similar problems are :
1. Initial lat., 40° 28' N. ; initial long., 73° 50' W. ; course,
119°; dist., 2924 miles. This would take the ship from
Sandy Hook to St. Vincent, Cape Verde Islands.
Ans. Final lat., 16° 50' N. ; final long., 25° 7' W.
2. Initial lat., 40° 10' N. ; initial long., 70° 0' W. ; course,
75° ; dist., 2606 miles. This would take the ship from Nan-
tucket Lightship to Fastnet, the nearest point of the Irish
coast.
Ans. Final lat., 51° 24' N. ; final long., 9° 37' W.
Before proceeding to our second fundamental problem
(p. 8), it will be well to explain briefly two further points
of interest. The first of these relates to the method of desig-
nating a ship's course. We have hitherto supposed it to
be measured in degrees, from the north, around by way of
the east, through the south and west, and so back to the
north again. This is the best way to count courses, and is
the way now in use in the United States Navy. Since a
whole circle contains 360°, it follows that courses may con-
tain any number of degrees from 0° to 360°.
But there is another quite convenient, although older, way
of designating courses, in which a 60° course, for instance, is
written N. 60° E., showing that the ship must be steered 60°
east of north. In a similar way, a 120° course is written
S. 60° E., showing that the helmsman should head her 60°
east of south, which would be the same as 30° south of east,
or 120° from the north toward the south by way of east.
The second further point of interest has to do with the
relation between Tables 1 and 2. It is possible to avoid
entirely the use of Table 2, and to transform longitude differ-
ences into departures, and vice versa, by means of Table 1
20 NAVIGATION
alone. It so happens that the relation between these two,
for any given middle latitude, as, for instance, 29°, is iden-
tical with the relation between distance and latitude difference
in Table 1 for the course 29°. In other words, if we have
given a middle latitude and a longitude difference, and wish
to find the departure, we :
Call the middle latitude a course, and
Call the longitude difference a distance ;
Then, corresponding to that course and distance, find from
Table 1 the tabular latitude difference, and it will be
the required departure. The same process can also be
reversed, so as to find the longitude difference from the
departure.
While this method with Table 1 is quite correct, we believe
beginners (at least) will find the use of Table 2 advantageous
in the solution of these problems, especially when the middle
latitude is not very great.
Coming now to our second fundamental problem of dead
reckoning, let us suppose a ship is required to proceed from
the initial lat. 42° 11' N. and long. 59° 28' W. to a final
lat. 42° 59' N. and long. 58° 33' W. We are to find the course
she must steer, and the distance she must run.
We have at once the latitude difference of 0° 48', or 48 miles,
and the middle latitude 42° 35', or nearest whole degree of mid-
dle latitude, 43°. The longitude difference is 55' ; and with this
we find from Table 2 the correction 14.8 in the 43° column
of middle latitude. Remembering that this time we are
transforming a longitude difference into departure, and con-
sequently do not need to use the factor at the foot of the
column, we subtract this correction (14.8) from the longi-
tude difference (55') and obtain the departure as 40.2 miles.
Next we proceed to Table 1, to find the course and distance
corresponding to lat. 48, dep. 40.2. To do this, we must
find a place in Table 1 where this particular latitude and
departure appear side by side. If this pair of numbers
DEAD RECKONING WITHOUT LOGARITHMS 21
cannot be found (exactly) side by side, we must take the
pair which come nearest to them : in this case such a pair
of numbers is found in the 40° course column, opposite dist.
63. So it appears that the ship must steer on a 40° course
a distance of 63 miles, to proceed from the given initial to
the given final latitude and longitude. This problem is the
direct converse of the one first solved (pp. 15, 17).
As a second example, let us now calculate the course and
distance from Sandy Hook, lat. 40° 28' N. ; long. 73° 50' W.,
to St. Vincent, lat. 16° 50' N. ; long. 25° 7' W. We have,
by subtraction, lat. diff. = 23° 38' = 1418' = 1418 miles;
long. diff. = 48° 43' = 2923'.
This 2923' must be turned into a departure, the middle
latitude being 28° 39', or, to the nearest whole degree, 29°.
Turning to the column of Table 2 which belongs to 29° of
middle latitude, we find the correction for 2923' of longitude
difference thus :
Tabular number for 900 = 113.0,
which being multiplied by 3, gives :
Tabular number for 2700 = 339.0
Also, tabular number for 200 = 25.1
Tabular number for 23 = 2.9
Sums, tabular number for 2923 = 367.0
This must be subtracted from the longitude difference, and
so we get :
dep. = 2923 - 367.0 = 2556 miles.
We have now to seek a place in Table 1 where lat. 1418 and
dep. 2556 appear side by side. No traverse tables are suffi-
ciently extended to contain these large numbers, but we
can at once obtain an approximate answer to the problem
by dividing both numbers by 100. This reduces them to
lat. 14.2, dep. 25.6 ; and the nearest numbers to these which
can be found side by side in Table 1 are in the column belong-
ing to course 119° and opposite dist. 29. This course (119°)
is the same as would have been obtained if we had not been
22 NAVIGATION
forced to divide our latitude and departure by 100, to bring
them within the range of Table 1. But the dist. 29 must
now be multiplied by 100, to remove the effect of our former
division of latitude and departure by 100. Thus we have
the closely approximate information that the course and
distance from Sandy Hook to St. Vincent are 119° and 2900
miles. The same problem (p. 19), when taken in its inverse
form, starts with the numbers 119° and 2924 miles.
In discussing such a problem, many beginners have dif-
ficulty in choosing correctly the course number (119°) from
the four (61°, 119°, 241°, 299°) to be found at the foot of
the same column of Table 1. This choice is easily made with
the help of our knowledge of elementary geography, or with
any rough chart or map. From these, we know that St.
Vincent is south and east of Sandy Hook, and the only one
of the four possible courses that will carry a ship south and
east is course 119°. The same course might be written in
the other notation (p. 19) S. 61° E., which possibly makes
the actual direction to be steered a little easier to under-
stand.
The above result is approximate only, but higher accuracy
is seldom required. When desired, it can be obtained by
certain kinds of interpolations (p. 12) ; but these are always
unsatisfactory, especially as complete precision can always
be easily had by the use of logarithms, as explained in the
next chapter.
CHAPTER III
DEAD RECKONING WITH LOGARITHMS
SINCE the publication in 1876 of Kelvin's tables for
facilitating Sumner's method, it has been possible to navi-
gate in the most approved way without using logarithms or
trigonometry. Those who desire to study the subject in
this manner may do so by simply omitting those parts of
the book in which logarithmic or trigonometric formulas
and calculations occur. But this method of study is not
recommended, except perhaps for a first reading; for a
knowledge of logarithmic processes always affords a most
desirable check on the accuracy of the other method, and
so makes for safety of the ship and peace of mind of the
navigator.
Proceeding, then, with the subject of logarithms, we may
define them as a mathematical device for facilitating calcula-
tions. They are merely numbers; but they are numbers
having this peculiarity : every logarithmic number belongs
to some ordinary number (like 1, 2, 3, 27, 800, etc.), and
belongs to it, alone. Its logarithm belongs to the number as
a man's shadow belongs to the man.
For our present purpose it is unnecessary to enter into the
theory of logarithms ; we shall explain only the methods of
using them in practice. Logarithms (abbreviated "log")
always consist of two parts, a "whole number" part and a
"decimal" part. Thus, 3.30103 is a logarithm, of which
the whole number part is 3, and the decimal part .30103.
The whole number part may even be zero : thus, 0.30103
is also a logarithm. The decimal part of the logarithm
is found from a table of logarithms, such as our Table 3
23
24 NAVIGATION
(p. 178) ; but the whole number part is found by an inspec-
tion of the number to which the logarithm belongs.
We shall hereafter, to save space, always write "log 26"
in place of "the logarithm belonging to 26": and, with
the help of this abbreviation, we may now write the follow-
ing tabular statement, which is fundamental in the matter
of logarithms :
log 1 = 0.00000, log 1000 = 3.00000,
log 10 = 1.00000, log 10000 = 4.00000,
log 100 = 2.00000, log 100000 = 5.00000, etc.
In other words, for these particular numbers, all "mul-
tiples" of 10, the decimal part of the log is zero. For
numbers intermediate between 1 and 10, the whole number
part of the log is 0, and the decimal part lies between
.00000 and .99999. For those between 10 and 100 the whole
number part is 1, and the decimal part again lies between
.00000 and .99999.
The general rule is : the whole number part of a log is
one less than the number of figures or "digits" in the number
to which the log belongs. Thus, the number 26 has two
digits : the whole number part of its log is 1. The number
2678 has four digits : the whole number part of its log is
therefore 3.
If a number is itself partly decimal, we count only the
number of digits to the left of the decimal point for the pur-
poses of the present rule. Thus, 26.78 has two digits only;
2.678 has 'one; 267.8 has three, etc.
If, on the other hand, a number is wholly decimal, as
0.2678, the whole number part of its logarithm should be
"negative," or minus, i.e. less than 0; and it will be one
greater than the number of zeros immediately following the
decimal point in the number. According to this, the whole
number part of log 0.2678 should be — 1, because this
number has no zeros immediately following the decimal
point. But as these negative whole number parts are
very inconvenient in actual work, it is customary to increase
DEAD RECKONING WITH LOGARITHMS 25
all logs of decimal numbers arbitrarily by 10, which will
avoid the negative sign. This arbitrary increase is always
corrected again in the further or final procedure, so that it
cannot possibly introduce error into the work.
In the case of log 0.2678, the arbitrary increase of 10
changes the — 1 to + 9 l ; and so 9 would be the whole
number part of log 0.2678. Similarly, log 0.002678 would
have 7 for its whole number part, because there are two zeros
after the decimal point. This would make the whole number
part of the log — 3, which, being increased by 10, gives + 7.
In general, this matter of logs of wholly decimal numbers
may be summarized as follows :
log 0.1 =9.00000, log 0.0001 =6.00000,
log 0.01 =8.00000, log 0.00001 =5.00000,
log 0.001 = 7.00000, log 0.000001 = 4.00000, etc.
In all these cases the decimal part of the log is zero :
and if the number lies, for instance, between 0.1 and 0.01,
the whole number part of the log will be 8, and the decimal
part will lie between .00000 and .99999.
The decimal part in the log of any number is taken from
Table 3 without regard to the position of the decimal
point in the number itself. The numbers 0.2678, 0.002678,
26.78, 2.678, 267.8, and 2678 all have precisely the same
decimal part in their logs, so that such logs will differ in
their whole number parts only. We can at once obtain this
common decimal part from Table 3 (p. 181), where it is
found to be .42781. In looking up this log, we again use
(p. 11) a pair of arguments. The argument for the left-
hand column consists of the first three digits of 2678 (267) ;
and in selecting this argument we disregard any zeros that
may immediately follow the decimal point, if the number
is wholly decimal, like .002678. The other argument, in
the top horizontal line of the tabular page is 8, the right-
hand digit of the number 2678. In the horizontal line
1 According to Algebra, 9 is greater than - 1 by 10.
26 NAVIGATION
opposite 267, and in the column headed 8, appears 781 ; and
these are the last three digits of the required log (.42781).
The first two digits (.42) are common to a great many logs,
and are therefore only printed in the column headed 0.
The first two digits of every log are thus taken from the
zero column, regularly from the same horizontal line that
contains the last three digits of the log, or from some line
above it. Only when there is an asterisk printed in the table
with the last three digits do we make an exception, and take
the first two digits from tha line below the one containing the
last three. Thus the decimal part of log 2691 is .42991, but
the decimal part of log 2692 is .43008.
Having thus found the decimal part of log 2678 to be
.42781, and the number 2678 having four digits, the com-
plete
log 2678 = 3.42781 ;
and here the reader should once more note that all tabular
logs like .42781 are thus always decimals. The correspond-
ing logs for the other numbers given above are :
log 267.8 = 2.42781,
log 26.78 = 1.42781,
log 2.678 = 0.42781,
log 0.2678 = 9.42781,
log 0.002678 = 7.42781.
It is clear that Table 3 gives directly the decimal part of
the logs of all numbers containing four digits. If the number
contains less than four digits, as 26, we should look it up in
the table as if it were 2600. We should find 260 as the
argument in the left-hand column (p. 181) ; and in the
corresponding line, in the column headed 0 (the fourth digit
of 2600), is 41497. This is the decimal part, as usual, and
the complete
log 26 = 1.41497.
If, on the other hand, the number whose log is wanted
contains more than four digits, as 26782, it is necessary to
DEAD RECKONING WITH LOGARITHMS 27
resort to interpolation (p. 12). The number of digits being
here 5, the whole number part of the log is 4 (p. 24). The
decimal part of the log is to be found quite without regard
to decimal points (p. 25). It may therefore be taken
from Table 3 just as if we wanted log 2678.2 instead of 26782.
Now the table tells us (p. 181) :
decimal part of log 2678 = 42781,
decimal part of log 2679 = 42797.
The tabular difference (p. 12) of these two decimal parts
is 16. As 26782 may, for our present purpose, be regarded
as lying & of the way from 2678 to 2679, it follows that the
decimal part of log 26782 will lie ^ of the way from 42781
to 42797. Evidently, we must multiply the tabular differ-
ence 16 by -£$ (giving 3.2) to find how much larger the decimal
part of log 26782 is than the decimal part of log 2678.
This 3.2 (or 3, in round numbers) must then be added to
42781 ; and we have, as the result of this interpolation :
decimal part of log 26782 = .42784.
As we have just found the whole number part to be 4,
we have for the complete :
log 26782 = 4.42784.
This whole process of interpolation may perhaps be more
clearly understood if we repeat (p. 10) that all tables furnish
tabular numbers corresponding to given arguments. In-
terpolation is necessary when the given arguments are not
to be found in the argument part of the table, but fall
between two of the tabular arguments. Then we obtain
by subtraction the difference between the given argument
and the nearest smaller argument contained in the table.
This difference is the "argument difference" (abbreviated,
arg. diff.), and it should be expressed as a decimal fraction
of the interval between two successive arguments (cf. •£$,
above). The tabular difference (tab. diff.) between two
successive tabular numbers being also obtained by subtrac-
28 NAVIGATION
tion, we have only to multiply the tabular difference by the
argument difference to find the "interpolation difference"
(int. diff.)- This is then added 1 to the proper tabular
number (belonging to the above-mentioned nearest argu-
ment given in the table) to obtain the tabular number re-
quired.
The multiplication of the tabular difference by the argu-
ment difference is facilitated by certain little auxiliary mul-
tiplication tables (called tables of "proportional parts")
printed in the margins of many mathematical tables. In
the example given above, the tabular difference was 16 ; and
Table 3 contains on the proper page (p. 181) a proportional
part table headed with this same number 16 ; and it shows
that for an argument difference .2, and tabular difference 16,
the interpolation difference is 3.2, just as we found above.
Other examples of logarithms are :
log 427 = 2.63043, log 42765 = 4.63109,
log 4276 = 3.63104, log 282374 = 5.45082,
log 0.4276 = 9.63104, log 2 = 0.30103,
log 0.42765 = 9.63109, log .0027 = 7.43136.
The above considerations are preparatory only to the
actual use of Table 3 ; and they are not yet quite complete.
For it is still necessary to explain the inverse use (p. 12) of
the table, or, in other words, the finding of the number to
which a given log belongs. Thus, if the given log were
3.42781, we should begin by looking up its decimal part
among the logs in the table. Finding it there, we take out
the number to which it belongs, 2678. We then put in the
decimal point according to the whole number part of the log.
This being 3, we know (p. 24) that the number required must
contain 4 digits. Therefore :
number to which the log 3.42781 belongs = 2678.
1 Except when a glance at the table shows that the tabular num-
bers are growing smaller, in which case the interpolation difference
must be subtracted. This never occurs in Table 3, but happens fre-
quently in Table 4.
DEAD RECKONING WITH LOGARITHMS 29
If the given log had been 2.42781, the table would furnish
the same number 2678, but the decimal point would be
differently located. Because the whole number part of the
given log is now 2, we know that the number to which it
belongs has three digits, and so :
number to which the log 2.42781 belongs = 267.8.
When the given log is not to be found in the table exactly,
a process of inverse interpolation is, of course, necessary.
Thus, if the given log is 4.42784, we look for its decimal
part in the table, and find it lies between
42781, which belongs to the number 2678, and
42797, which belongs to the number 2679.
The decimal part of the given log being 42784 is greater by
3 than the nearest tabular number 42781. This 3 is there-
fore the interpolation difference. The tabular difference is
16, obtained by subtraction between 42781 and 42797. We
now divide the interpolation difference by the tabular dif-
ference, which gives .2 (^ = 0.2, in round numbers). This
.2 is the argument difference, and therefore the complete
number belonging to the decimal part of 'the log (42784)
is 26782. The whole number part of the given log
being 4, the required number must have 5 digits, and will
therefore be 26782. Had the given log been 2.42784, we
should have arrived at the number 26782 in just the same
way; but we should locate the decimal point differently.
The whole number part of the log being now 2, there should
be only 3 digits in the number, and we should have :
number to which the log 2.42784 belongs = 267.82.
Other similar examples are :
log = 2.71828, corresponding number = 522.73,
log = 4.26323, corresponding number = 18333,
log = 9.26323, corresponding number = 0.18333,
log = 0.21000, corresponding number = 1.6218.
The reader will perceive, from a consideration of these
interpolated numbers, that work with logarithms is never
30 NAVIGATION
exact, absolutely. This is inherent hi the nature of our
log tables, which really contain only the decimal parts of the
logs carried out to five places of decimals. Further decimals
of course exist, but are here omitted, because five places
always give sufficient accuracy for navigation calculations.
The simplest calculations which are facilitated by loga-
rithms are the ordinary arithmetical processes of multi-
plication and division. These processes can be turned into
addition and subtraction by the use of the following
principle :
The log of a product is equal to the sum of the logs of the
factors.
According to this principle, if we wish to multiply a series
of factors, we simply add their logs. The sum is then a log
and the number to which this log belongs is the product of the
series of factors. Suppose, for instance, we wish to multiply
the factors 2, 3, and 4. The product should be 24. Proceed-
ing with logs, we have from Table 3 :
log 2 = 0.30103,
log 3 = 0.47712,
log 4 = 0.60206,
log product = sum = 1.38021,
and the number to which the log. 1.38021 belongs is, accord-
ing to Table 3, 24.00, the correct product.
It is evident that the use of the log table is here of no
advantage, because the factors are very small : but when
large numbers are to be multiplied the advantage is very
great.
Taking now a similar simple example of division, let us
divide 6 by 3. In division, evidently, we must subtract
the log of the divisor from the log of the dividend, to obtain
the log of the quotient. We have
log 6 = 0.77815,
log 3 = 0.47712,
log | = difference = 0.30103,
DEAD RECKONING WITH LOGARITHMS 31
and the number to which the log 0.30103 belongs is 2.000,
the correct quotient. Other examples are :
2.426 X 42.78 X 17.26 = 1791 .3,
6.242 X 87.24 x 62.71 = 34149,
ff|= 1.6234,
24 = °'75'
In the last example, we have
log 18 = 1.25527,
log 24 = 1.38021.
The subtraction would lead to a negative log because 1.38021
is larger than 1.25527. Therefore we arbitrarily increase
1.25527 by 10, giving 11.25527, and then the subtraction
gives
log quotient = 9.87506,
which is the log belonging to the number 0.75, the correct
quotient.
We come now to the solution of the two fundamental
problems of dead reckoning (pp. 8, 10) by means of logs.
For this purpose we must use our Table 4, in connection with
Table 3. Table 4 is called a trigonometric log table and
the tabular numbers in it are certain logs known as :
sine, abbreviated sin, cotangent, abbreviated cot,
cosine, abbreviated cos, secant, abbreviated sec,
tangent, abbreviated tan, cosecant, abbreviated esc.
It is not our purpose to consider the theory of trigonom-
etry, but it is necessary for the reader to have
some understanding of its practical applica-
tions. If we have a triangle QPY (fig. 6), we
notice that it is made up of six "parts," the
three sides and the three angles. Now it is a
fact that if we know any three of these six y
parts, we can calculate the other three parts, FIG. 6.— Trigo-
provided one of the known parts is a side.
Trigonometry is the branch of mathematics which enables us
32 NAVIGATION
to do this, and the triangle QPY is the very triangle which
occurs in the two problems of dead reckoning,
In trigonometry, every angle has belonging to it a sin,
cos, etc., just as every number has its log. These sines,
etc., can be taken out of Table 4 by means of a pair of argu-
ments in the usual way. The two arguments are the number
of degrees and the number of minutes in the angle (p. 9).
The number of degrees is found in Table 4 at the top or bottom
of the page, and the number of minutes in the right-hand or
left-hand column. Each page (as, for instance, p. 229) has
eight degree numbers, four, 33°, (213°), (326°), and 146° at
the top, and four, 123°, (303°), (236°), and 56° at the bottom.
The proper sines, etc., for all these degrees appear on the
same page (p. 229). When the degree number is at the top
or bottom of the left-hand column 33°, (213°), (303°), and
123°, the minutes must be taken from the left-hand column.
But when the number of degrees is at the top or bottom of the
right-hand column 146°, (326°), (236°), and 56°, the minutes
must come from the right-hand column. And when the
number of degrees comes from the top of the page, we must
look for the proper sine, etc., in a column having the word
sin, etc., at the top. But when the degree number comes
from the bottom of the page, the sine, etc., will be taken
from a column having the word sin, etc., at the bottom.
Thus (p. 229) :
sin 33° 26' = sin 146° 34' = cos 56° 34' = cos 123° 26' = 9.74113.
In this way, sines, tangents, etc., can be taken from
Table 4. Examples are :
sin 28° 32' = 9.67913, cot 117° 10' = 9.71028,
cos 66° 14' = 9.60532, sec 12° 40' = 0.01070,
tan 128° 28' = 0.09991, esc 111° 11' = 0.03038.
These sines, etc., are really all logs. When the whole num-
ber part is 9, it indicates that the log belongs to a number
which is wholly decimal (see p. 24), and that the log has
been arbitrarily increased by 10.
DEAD RECKONING WITH LOGARITHMS 33
Of course these trigonometric tables can also be used in
the inverse manner. Thus, to find the angle corresponding
to the sin 9.28190, we turn to p. 207, and finding 9.28190 in
the sin column, we see that the corresponding angle is
either 11° 2', 191° 2', 168° 58', or 348° 58'. When the sin,
etc., cannot be found in the table exactly, we may always
take the nearest one : interpolation is never practically
necessary in using the trigonometric tables in navigation.
Examples are :
sec = 0.17177, angle = 47° 40', 227° 40', 132° 20', or 312° 20',
tan = 0.17177, angle = 56° 3', 236° 3', 123° 57', or 303° 57',
sin = 9.17177, angle = 8° 32', 188° 32', 171° 28', or 351° 28',
cos = 9.17177, angle = 81° 28', 261° 28', 98° 32', or 278° 32',
esc = 0.17177, angle = 42° 20', 222° 20', 137° 40', or 317° 40',
cot = 0.17177, angle = 33° 57', 213° 57', 146° 3'i or 326° 3'.
Having thus explained the use of Table 4, we shall now
apply it to the two problems of dead reckoning. These
problems are :
1. To find latitude difference and departure from course
and distance ;
2. To find course and distance from latitude difference
and departure.
These problems are solved by means of the following
formulas, in which the letter C represents the course angle :
n . f log lat. diff. = log dist. + cos C,
"J [ log dep. = log dist. + sin C.
I tan C = log dep. — log lat. diff.,
* j log dist. = log dep. — sin C.
Sometimes it is preferable to find the distance from the
latitude difference instead of the departure. We then use
the following modification of formula (2) :
(2') log dist. = log lat. diff. - cos C.
Let us now solve with these formulas our former problem
(p. 18), in which a ship traveled 1377 miles on a course of
166°. Applying formula (1) above, we have :
34 NAVIGATION
log dist. (1377) =3.13893 log dist. (1377) =3.13893
cos C (166°) = 9.98690 sin C (166°) = 9.38368
sum = log lat. diff. = 3.12583 x sum = log dep. = 2.52261 1
corresponding lat. diff. = 1336.1 corresponding dep. = 333.1
These corresponding latitude difference and departure
agree very closely with the results already found (p. 18)
from Table 1.
If the departure and latitude difference were given, we
could find the course and distance by means of formula (2)-
In the present case we have :
log dep. (333.1) =2.52261 log dep. (333.1) =2.52261
log lat. diff. (1336.1) = 3.12583 sin C (166°) = 9.38368
by subtraction, tan C = 9.3967S2 by subtraction, log dist. = 3.138933
corresponding C = 166° corresponding dist. = 1377
These numbers, 166° and 1377 miles, are the same numbers
with which we began this calculation ; so it is clear that the
log method of calculation agrees with the traverse table
method. For accuracy the log method is superior.
The transformations of departure into longitude differ-
ence, and vice versa, are accomplished logarithmically with
the following formulas :
(3) log long. diff. = log dep.— cos middle lat.
(4) log dep. = log long. diff. + cos middle lat.
Thus the longitude difference corresponding to dep. 333.1
would be calculated by formula (3) as follows :
log dep. (333.1) =2.52261
cos mid. lat. (29° 16'; p. 18) = 9.94069
by subtraction, log long. diff. = 2.58192
corresponding long. diff. = 381 '.9 = 6° 21 '.9.
1 These numbers have been diminished by 10, to allow for the fact
that both cos C and sin C have been arbitrarily increased by 10 (p.
32; cf. also p. 25).
2 This number has been increased by 10, and therefore is in accord
with the usual practice of avoiding negative whole numbers in the
trigonometric Table 4.
3 This subtraction is correct, if we remember that the 9.38368 is
really too large by 10.
DEAD RECKONING WITH LOGARITHMS 35
This is in close accord with the result on p. 18, where
Table 2 gave 6° 20'. 5. The logarithmic method is again
the more precise, for it takes account of minutes in the course,
which were neglected on p. 18. But either result is accurate
enough for practical purposes.
Before finally leaving these problems of dead reckoning,
we shall explain briefly two additional methods of solving
them which differ from the method so far employed. These
two additional methods are called "Mercator sailing" and
"great circle sailing"; whereas, up to the present, we have
been using "middle latitude sailing," so named because
the middle latitude appears in the calculations.
Mercator sailing is based on a kind of chart first designed
by Gerhard Mercator, a sixteenth century geographer.
Such charts are still widely used for nautical purposes.
In calculations based on them, every parallel of latitude is
referred directly to the equator by means of a table of "merid-
ional parts." Our Table 5 is such a table, and it gives the
meridional part for every degree and minute of latitude
from the equator to 60°. These meridional parts are really
the distances from the equator to the several parallels of
latitude, such as they would appear on a Mercator chart
drawn to such a scale that 1' of longitude at the equator would
occupy one linear unit on the chart. Thus the meridional
part for lat. 40° is given in Table 5 as 2607.6. Suppose the
scale of the chart at the equator were 1 inch to the degree of
longitude. That would be •£$ inch to the minute. The dis-
tance on the chart from the equator to the 40° parallel of
latitude would then be 2607.6 X ^ inches = 43.46 inches.
It is needless to say that a chart on such a scale could not
show a very large part of the ocean on a single sheet.
Calculations by Mercator sailing are of course only made
when the distances involved are large and great accuracy is
required. It is therefore best to do them by means of
logarithms, although it is also possible to obtain Mercator
results from the traverse table . In such calculations we do not
36 NAVIGATION
use the latitude difference of ordinary middle latitude sailing.
In its place appears the "meridional latitude difference" (ab-
breviated mer. lat. diff .), defined as the difference between the
meridional parts (Table 5) belonging to the two latitudes
(initial and final) involved in the problem. With this defini-
tion in mind we may now give the Mercator formulas as
follows :
(5) log mer. lat. diff. = log long. diff. + cot C.
(6) log long. diff. = log mer. lat. diff. + tan C.
(7) tan C = log long. diff. - log mer. lat. diff.
Let us now apply these formulas to the problem of pp. 18
and 33, in which a ship starts from the initial lat. 40° 24' N. ;
long. 73° 58' W., and travels 1377 miles on a course, C,
of 166°. What final latitude and longitude does she at-
tain ? The latitude difference is found in the ordinary way
(p. 34), there being no special Mercator formula for it, and
comes out 1336.1 miles, or 1336M = 22° 16'. The final lati-
tude (p. 18) is therefore 40° 24' - 22° 16' = 18° 8'. Then,
from Table 5, we have :
for initial lat. 40° 24', mer. parts = 2638.9
for final lat. 18° 8', mer. parts = 1099.4
by subtraction,1 mer. lat. diff. = 1539.5
Now, applying formula (6), we have:
log mer. lat. diff. (1539.5) (Table 3, p. 179) = 3.18738
tan C (166°) (Table 4, p. 209) = 9.39677
by addition, log long. diff. = 2.58415
corresponding long. diff. (Table 3, p. 183) = 383'.8 = 6° 24'
The final longitude is therefore 73° 58' - 6° 24' = 67° 34' W.,
whereas we obtained before 67° 38' W. (p. 18).
Finally, we shall apply the Mercator method to the
example of p. 21. It is required to find the course and
distance from
Sandy Hook, lat. 40° 28' N. ; long. 73° 50' W. to
St. Vincent, lat. 16° 50' N. ; long. 25° 7' W.
1 If one latitude were in the southern hemisphere and the other
in the northern, we should add the meridional parts.
DEAD RECKONING WITH LOGARITHMS 37
We have from Table 5 :
for initial lat. 40° 28', mer. parts = 2644.2
for final lat. 16P 50', mer. parts = 1018.1
by subtraction, mer. lat. diff. = 1626.1
The longitude difference is found by subtraction to be
73° 50' - 25° T = 48° 43' = 2923'. Now applying formula
(7), we have :
log long. diff. (2923) (Table 3) = 3.46583
log mer. lat. diff. (1626) (Table 3)= 3.21112
by subtraction, tan C = 0.25471
and therefore (Table 4) C =• 119° 5'.
The distance is found in the ordinary way from the
latitude difference (not mer. lat. diff.) by means of formula
(20, P. 33.
The latitude difference is 40° 28'- 16° 50' = 23° 38' = 1418'.
Formula (2') then gives :
log lat. diff. (1418') (Table 3) = 3.15168
cos C (119° 5') (Table 4) = 9.686711
by subtraction, log dist. = 3.46497 1
corresponding dist. (Table 3) = 2917
Course 119° 5', distance 2917 miles is therefore the
solution by Mercator sailing. On p. 22, we obtained 119°
and 2900 miles; and on p. 19 we began with 119° and 2924
miles. The agreement is satisfactory.
Having thus briefly described Mercator sailing, we come
next to "great circle sailing." This is a method of determin-
ing the ship's course toward her port of destination in such a
way that the distance to be traveled will be as short as
possible. If the earth's surface were flat instead of spherical,
the shortest course would be a straight line, as used in plane
sailing; but on the sphere the shortest course is a curve
called a "great circle." Evidently, on all long voyages, the
great circle course is the most advantageous one; that
mariners do not more frequently use it is due to a peculiarity
of their charts.
1 This log is really too large by 10, so the subtraction is correct.
38 NAVIGATION
We cannot here enter into the details of chart "pro-
jections," as the theory of chart making is called. It is
sufficient to remark that a straight line drawn on the ordi-
nary nautical charts (which follow the Mercator system),
between any two ports, will not represent the shortest (or
great circle) course between them. On such a chart, the
great circle course between the two ports will appear to be
longer than the straight line course, although it is really
shorter. This accounts for the use of the longer Mercator
course by many navigators.
Now there is a kind of chart, called a "great circle sailing"
chart, on which straight lines between ports really represent
shortest (or great circle) courses. One would therefore
naturally suppose that mariners would entirely discontinue
the use of Mercator charts in favor of great circle charts.
But there is a reason for not doing this.
On Mercator charts, all terrestrial longitude meridians
are represented by parallel vertical straight lines. Conse-
quently, if we draw another straight line on 'the Mercator
chart joining two ports, that line will make the same course
angle (p. 10) with all the meridians. In this way, a navigator
can get from a Mercator chart, by simply drawing a straight
line, and quite without calculation, a course angle which will
carry him from one port to another. And because the course
angle so obtained is the same with respect to all meridians
to be crossed by the ship it follows that the voyage can be
completed (theoretically at least) from the one port to the
other with the great advantage of never changing the course
to be steered.
On the other hand, the great circle track makes a different
angle with every meridian it passes : so that the mariner
must make very frequent changes in the course angle to be
steered during the progress of a voyage. The simple
Mercator track, without change of course, is called a "rhumb
line" ; the serious objection to it is that it sometimes leads
to greatly (and unnecessarily) lengthened voyages.
DEAD RECKONING WITH LOGARITHMS 39
The final conclusion is that Mercator charts, on account of
their simplicity, are most convenient for short voyages, or
for parts of long voyages when the land is not far away.
But for shaping the main part of the course on a very long
voyage, great circle sailing charts are to be preferred.
At times, in order to avoid very high latitudes, or to round
some projecting point of land, navigators must substitute for
a single great circle track one "composed" of two or more
shorter arcs of great circles. This is called "composite"
sailing.
Finally, for the sake of completeness, we shall merely
mention two other kinds of sailing. " Parallel " sailing, which
is simply middle latitude sailing when the latitude difference
is zero; and "traverse" sailing, from which the traverse
table gets its name. This is also the same thing as middle
latitude sailing; but the special word "traverse" is used
when the ship changes her course frequently, perhaps even
during a single day. It is then possible to sum up the
result of all the short courses which together make up the
day's run. It is merely necessary to take from the traverse
table the latitude difference and departure for each short course
separately, and then to add 1 all the values of latitude differ-
ence for a "summed latitude difference," and all the values
of departure for a "summed departure." With these a
"composite course and distance" can be taken from the
traverse table, or calculated with logs, and these will repre-
sent the motion of the ship, just as if she had steered an
unchanged course during the entire day.
1 It is necessary to sum separately latitude differences represent-
ing northward motion of the ship and those representing southward
motion. The difference of the two sums is what we need to know.
The same is true of departures representing eastward and westward
motion of the ship.
CHAPTER IV
THE COMPASS
THE ship's course has been defined (p. 8) as the angle
between the north and the direction in which the ship is
sailing. To ascertain what this angle is, or, in other words,
to steer the ship, mariners use the compass. The dial (or
"card") of this instrument is divided, like any circle, into
360°. In the United States Navy these are numbered in
such a way (fig. 7) that 0° appears at the north, 90° at the
east, 180° at the south, and 270° at the west. The numbers
therefore increase in a "clockwise" direction. There are
also compasses in which the numbering begins with 0° at
both the north and south points, and increases to 90° at the
east and west points. But the United States Navy system
of numbering is to be preferred.
In addition to the above division and numbering, the dial
is also divided into 32 points (pp. 10, 15), each containing
ocn°
, or 11|°. These points are then further subdivided
o&
into quarter points, all of which is shown clearly in Fig. 7.
The naming of the points has not been done by chance,
but in accordance with a definite rule. The four principal,
or "cardinal," points are north, east, south, and west. The
remaining points are located by a continued process of
halving. Halfway between the cardinal points are the
"inter-cardinal" points; and each is named by combining
the names of the two cardinal points adjacent to it. Thus
northeast (abbreviated N.E.) is halfway between north
and east. Again halving and combining names, we get
points like E.N.E., S.S.E., etc. Still once more halving
completes the tally of 32 points : but a combination of
names would now be too complicated. However, since
40
THE COMPASS
41
each of these final points must necessarily be adjacent to a
cardinal or inter-cardinal point, they are named by simply
increasing the name of such adjacent cardinal or inter-
cardinal point. This is accomplished with the word "by."
FIG. 7. — Compass Card.
Thus we find, adjacent to N.E., the points N.E. byE., and
N.E. by N. In the light of the above, it is easy to "box"
the compass, as seamen say, or to name the 32 points in
order.
When the point system of division is used, and an accuracy
42 NAVIGATION
closer than a single point is required, the compass card is
still further subdivided into quarter points. In naming
these it is customary, in the United States Navy, to "box"
from N. and S. towards E. and W. Thus the space between
N.N.E. and N.E. byN. would be divided into four parts
thus : N.N.E.iE., N.N.E ^E., N.N.E.f E. But an excep-
tion is made to this last rule' in the case of quarter points
adjacent to a cardinal or inter-cardinal point. These last
are always put first in naming the quarter points. Thus,
between E. by N. and E., if we always boxed from N. towards
E., we should have : E. by N.|E., E. by N.^E., E. by N.f E.
But it is customary, because shorter, to name these quarter
points E.fN., E.£N., and E.|N.
Inside the "bowl" of the compass, and adjacent to the
card, a black line is marked on the bowl. This line is in
plain view of the steersman, through the glass cover of the
compass, and is called the "lubber line." When the ship
is headed in such a way that this line comes opposite N.E.,
for instance, on the card, the ship will be on a N.E. course,
which makes an angle of 45° with the north.
But would the ship really be traveling on a line making
a 45° angle with the geographic meridian, or direction of
the north pole of the earth? She would be doing so only
if the compass were absolutely correct. This is practically
the case with the "gyro-compass," a mechanical contrivance
now much used in the navy, but not the case with the ordi-
nary "magnetic" compass.
In Chapters II and III, concerning dead reckoning, we have
always used the word "course" as if all compasses were
absolutely correct. But since they are not correct, it is
now necessary to make allowance for their errors. In other
words, whenever we use a compass, we must first ascertain
the difference between the "true course" and the "compass
course." It must not be supposed from this statement that
a ship can be steered on two different courses at the same
moment. There is really only one direction along which
THE COMPASS 43
the ship is moving : but the angle between that direction
and the true north may be different from the angle between
it and the "compass north." It is the course measured
from the true north that must be used in all dead-reckoning
calculations, and that always results from such calculations :
but for steering the ship by means of a compass the steers-
man must be furnished with the course as measured from
the compass north. Therefore it is essential for the navigator
to know the difference between the two. This difference
is called the "error" of the compass.
Unfortunately, this error is made up of two parts. The
first, called "variation" of the compass, is due to peculiari-
ties in the earth's magnetism, and is quite different in dif-
ferent places on the earth. It also varies in different years
at the same place. But at any one time, all ships in the
same part of the ocean will have the same variation.
The mariner can always ascertain how great the varia-
tion is in his part of the ocean, because it is always marked
on his chart. Certain curved lines are drawn on the chart ;
and if the ship is located on or near a line marked "varia-
tion 10°," for instance, it follows that the navigator must
on that day allow for 10° of variation. It is also important
to take into consideration possible changes in the variation.
Sometimes the annual change is marked on the chart; if
not, it is important to use a chart of recent date.
The second part of the error is called "deviation" and is
due to peculiarities in the magnetism always developed in
the metallic parts of the ship itself. It is different in dif-
ferent ships, even in the same part of the ocean, and is even
different in the same ship, when she is headed on different
courses. Methods have been invented for "compensating"
marine compasses, so as to remove the effects of deviation,
and these methods are quite effective. But even when
they are used, it is necessary, before beginning a long voyage,
to have a "compass adjuster" visit the ship. He will then
"swing" the ship on a number of different courses, and
44 NAVIGATION
adjust the compass so that it will be as nearly correct as pos-
sible. Finally, he will determine, by means of astronomic or
other observations, just what the remaining compass devia-
tion is on all the various courses, and give the navigator a
table of these remaining deviations. This table must be taken
into account in "shaping" the ship's course during the
voyage. The navigator must also, from time to time, check
these tabular deviations while at sea by means of astronomic
observations of his own, to take care of possible changes.
Such astronomic observations are made with an instru-
ment (the "azimuth circle"), which can be attached to the
compass, and with which the "compass bearing" of the
sun or any other object can be observed. The compass
bearing is simply the compass direction of the object, as
seen from the ship ; or the compass course on which the ship
would be steered, if she were moving directly toward the
object. When the sun is used, its true bearing, measured
from the true north, can be taken from astronomic tables
which will be explained later; and it is called the sun's
"azimuth." A comparison of this true bearing with that
measured on the compass with the azimuth circle then makes
the compass error known.
When it is not convenient to observe the sun, it is possible to
substitute observations of a distant well-defined terrestrial ob-
ject, whose true bearing can be measured on a chart for com-
parison with various compass bearings observed while the ship
is being swung. Another method is to set up a compass on
shore, away from any iron or steel, and use it to determine
the bearing of the distant object. And there is still another
method, if the above compass and the ship's compass are inter-
visible. For the bearing of each may then be taken from the
other, and these should differ by exactly 180°. If they do not,
the variation from 180° must be due to deviation on board.
The "pelorus" is another instrument which may at times
replace the azimuth circle. It is located anywhere on the
ship, at a convenient point for observation, and not neces-
THE COMPASS
45
sarily close to the compass. It has a "dummy card" and a
lubber line. The dummy card can be turned until the
lubber line indicates the same course as the real compass.
Observations of bearings with the pelorus will then obviously
be the same as if made on the compass with the azimuth circle.
The advantage of the pelorus is that it can be used anywhere
on board, while the compass must be kept constantly in the
exact place where it was "adjusted" before leaving port.
The error thus determined astronomically or otherwise
is the sum of the variation and deviation. If we indicate
by E the total compass error in that place, at that time, on
that ship, and on that course ; by D the deviation similarly
described ; by V the variation at that time and in that place ;
and if all three are counted from 0° in the usual direction
around the compass card, then
we have the formula :
(1) E = V + D.
By counting in the usual direc-
tion, we mean counting from the
north around to the east, as all
courses are counted (p. 19) ; so
that a compass error of 10°. for
instance, would mean that the
compass north pointed 10° east
of the true north, or had a true
bearing of N. 10° E. (p. 19).
This is shown in Fig. 8, which
also shows the ship's course,
counted in the same way.
It is clear from the figure that if we now indicate :
by C, the ship's compass course,
by T, the ship's true course,
by E, the compass error,
we shall have the formula :
FIG. 8. — Compass Error.
(2)
= C + E.
46 NAVIGATION
The simple formulas (1) and (2) enable the navigator to
make all necessary compass calculations. The following
are examples.
Suppose, for instance, that the error E has been deter-
mined by observation, and the variation V taken from the
chart. Formula (1) then makes it possible to calculate
the deviation D. For the formula shows that E is the sum
of V and D ; and so D must be the difference of E and V,
or: D = E - V.
Thus the deviation D becomes known, as a check on the
compass adjuster's work, and, while this value of D is cor-
rect only for the particular course on which the ship was
headed at the time the observation was made, yet that
course is the very one for which it is especially important
to have correct information.
Again, suppose dead-reckoning calculations show that the
ship is to sail on a 40° course. These calculations always
furnish the true course (p. 43) so that T = 40°. The
variation being known from the chart, and the deviation
from the adjuster's table, we know from (1) E = V + D.
Then from (2) we see that C = T — E, which gives the
compass course. Let us suppose in the present case, that
V was 9°, D 1° ; then E = V + D = 9° + 1° = 10° ; and
since T = 40°, C = T - E = 40° - 10° = 30° ; and the
helmsman would be directed to steer a 30° course by com-
pass.
If, in Fig. 8, the compass north happened to be 10° on the
left side of the true north, instead of the right, the error E
would be 350°, instead of 10° (see also fig. 7, p. 41). This
might be made up of a variation V of 349° and a deviation
D of 1°, as before. If the true course is again to be 40°,
the compass course would be 40° — 350°, according to the
formula C = T — E. This subtraction being impossible,
we increase the 40° by a complete circumference of 360°j
which is always permissible, and then have :
THE COMPASS 47
C = 360° + 40° - 350° = 50°.
The ship would be steered on a compass course of 50°.
An alternative way to take care of errors, variations,
and deviations on the left side of the true north is to mark
them with the negative or minus sign. Instead of calling
V 349°, we might call it — 11°. This is really the best way,
and leads to the same result as before, if we remember that
the subtraction of a minus quantity is always equivalent to
an addition. In the example just given, calling V — 11°,
instead of 349°, we should have : E = F + D = - 11° +
1° = - 10°; and C = T - E = 40° - (- 10°) = 50°, the
same compass course as before.
An older way of designating variations, deviations, and errors
is to call them east when the compass north points to the
right of the true north, and west when it points to the left
of the true north. This method leads to the necessity of
providing various rules or diagrams with which to make
compass calculations. We think the best way to avoid
error (and such errors may lose ships and lives) is to use the
method here given with its two simple formulas. When
some other designation of the error, or some other method
of numbering the card, is demanded by a captain, it is always
possible to conform to that demand, but also to translate
every problem into our method "(in imagination at least)
as a check against mistake.
The following is an example of a compass adjuster's "devia-
tion table," taken from Bowditch's " Navigator " (1916
edition). The deviations are set down in degrees and tenths
of a degree, instead of degrees and minutes, for convenience
in the further calculations. The ship was swung so that
her head bore successively around the horizon, and obser-
vations were made at intervals of 15°. This is a smaller
interval than is usually necessary ; and the deviations in the
table are much larger than commonly occur in a modern
well-compensated compass.
48
NAVIGATION
DEVIATION TABLE
BEARING
BEARING
BEARING
BEARING
OF SHIP'S
DEVIA-
OP SHIP'S
DEVIA-
OF SHIP'S
DEVIA-
OF SHIP'S
DEVIA-
HEAD BY
TION
HEAD BY
TION
HEAD BY
TION
HEAD BY
TION
COMPASS
COMPASS
COMPASS
COMPASS
o
o
0
o
o
o
o
o
0
- 15.5
90
- 9.1
180
+ 17.9
270
+ 9.9
15
- 14.9
105
-9.0
195
+ 23.8
285
+ 1.9
30
- 13.3
120
- 7.8
210
+ 27.1
300
- 4.2
45
- 11.3
135
- 5.9
225
+ 25.6
315
- 10.3
60
- 10.0
150
-2.3
240
+ 22.0
330
- 13.6
75
- 9.7
165
+ 8.5
255
+ 15.9
345
- 16.0
To illustrate the use of this table, let us suppose the ship
to be sailing on a compass course of 165°, in a part of the
ocean where the variation is + 10°, or 10° E. Using formula
(1) (p. 45), and finding from our table that the deviation D
for 165° is + 8°.5, we have the compass error E = V + D
= + 10° + 8°.5 = + 18°.5. By formula (2) (p. 45) the true
course of the ship is T = C + E = 165° + 18°.5 = 183°.5.
We should use this true course 183°.5 in calculating later
the ship's position by dead reckoning (p. 10).
If the compass variation were everywhere the same, it
would be more convenient to have a table of compass errors,
instead of a table of deviations ; but because the variation,
as given on the chart, varies greatly, the table must be
specially made for deviations only.
Equally important with the above use of our deviation
table is its inverse use. When the navigator has calculated
by dead reckoning the course he must steer, that course,
as it comes from the calculations, will be a true course (p.
43) ; and it is necessary to turn it into a compass course for
the use of the steersman.
To do this we must know the deviation ; and we cannot
get it directly from the deviation table above, because the
use of that table presupposes a knowledge of the compass
course, the very thing we are trying to find. The best
THE COMPASS
49
way to avoid this difficulty is to imagine the deviation to be
non-existent, for the moment, and to make use of the "mag-
netic course," defined as the course which would be indi-
cated by the compass, if deviation were thus totally absent.
Under these circumstances, formula (1) gives E = V, since
D = 0 ; and if we designate the magnetic course by M , we
may write, in place of formula (2) (p. 45) :
(3) M=T-V.
Let us suppose a case in which the variation is + 10°, and
the desired true course of the ship 175°. Then the magnetic
course, allowing for variation only, will be, by formula (3) :
M = T - V = 175° - 10° = 165°.
This course is not really a compass course, because no
account has yet been taken of the deviation. Nor can we
yet find the deviation directly from the deviation table,
because in that table we must still know the compass course
to use as the argument (p. 10), whereas we know as yet only
the magnetic course. Therefore navigators should always
request the compass adjuster to furnish a "second deviation
table," in which the argument is the magnetic course, in-
stead of the compass course. Such a second table can al-
ways be calculated from the other. We here give one that
has been calculated from the table on the preceding page.
SECOND DEVIATION TABLE
MAG-
MAG-
MAG-
MAG-
NETIC
NETIC
NETIC
NETIC
BEARING
DEVIA-
BEARING
DEVIA-
BEARING
DEVIA-
BEARING
DEVIA-
or SHIP'S
TION
OF SHIP'S
TION
OF SHIP'S
TION
OF SHIP'S
TION
HEAD
HEAD
HEAD
HEAD
0
0
0
o
o
o
0
o
0
- 14.9
90
-9.0
180
+ 11.0
270
+ 16.5
15
- 13.4
105
- 8.4
195
+ 16.9
'285
+ 4.1
30
- 11.7
120
- 6.9
210
+ 21.3
300
- 7.1
45
- 10.4
135
' -4.8
225
+ 24.9
315
- 13.2
60 -
- 9.8
150
- 1.4
240
+ 26.8
330
- 15.7
75
- 9.3
165
+ 5.0
255
+ 24.1
345
- 15.5
50 NAVIGATION
We also add as an example the calculation of one number
in the second table from those given in the first. We shall
find the deviation corresponding to the magnetic course
165° ; and we do it by a kind of interpolation (p. 12). From
the first table we have the deviation — 2°.3 for the compass
course 150°. Since the deviation is the only difference
between compass and magnetic courses, it follows that
150° — 2°.3, or 147°.7 magnetic, corresponds to 150° by com-
pass. Similarly, 173°.5 magnetic corresponds to 165° by
compass.
The magnetic course 165° for which we are making the
calculation falls between 147°.7 and 173°.5, and exceeds
the smaller of the two by 17°.3. The whole difference be-
tween 147°.7 and 173°.5 is 25°.8. Similarly, the whole dif-
ference between the two compass courses involved is 15°.
Therefore we may write the proportion :
25°.8 : 15° = 17°.3 : x°,
where x is the excess over 150° of the compass course corre-
sponding to 165° magnetic.
Solving this proportion by the ordinary rules of arithmetic,
we have :
= 15 X 17.3 = 1QO 0
25.8
The compass course belonging to 165° magnetic is there-
fore 150° + 10°.0 = 160°.0. The corresponding deviation
is 165° - 160°.0 = + 5°.0,1 which is therefore the deviation
for 165° magnetic, and appears as such in the second table.
This entire table can be computed from the first table in an
hour.
Sometimes the second deviation table gives compass courses
instead of deviations. It is then often called a " table of
1 A comparison of formulas (1), (2), and (3) shows that
D = M — C ; so that the deviation is obtained by subtracting the
compass course from the magnetic course. This is also evident
from the definition of a magnetic course (p. 49).
THE COMPASS 51
steering courses " ; and in the example just calculated it
would give the compass or steering course 160° for the mag-
netic course 165°, instead of giving the deviation + 5°.
We shall still further illustrate this important matter by
an example, supposed to occur on board a ship for which
our two deviation tables hold good.
What is the compass course to be given the helmsman at
Sandy Hook, on a voyage to St. Vincent?
We have already found, from dead-reckoning calculations
(p. 22) the course 119°. Being the result of a dead-reckon-
ing calculation, this is a true course. The track chart of
the north Atlantic gives the variation at Sandy Hook as
10° W., or - 10°. The true course being 119°, we get the
magnetic course, allowing for variation only, by formula (3),
M = T - V = 119° -(- 10°) = 129°. The second devia-
tion table shows that :
for magnetic course 120°, the deviation is — 6°.9, and
for magnetic course 135°, the deviation is — 4°.8.
Magnetic course 129° falls between 120° and 135°, so that
an interpolation (to be extremely exact) between — 6°.9
and — 4°. 8 makes the deviation for magnetic course 129°
come out — 5°.6. Formulas (1) and (2) now give :
Error =E = V+D = -W°- 5°.6 = - 15°.6
Compass course = C = T-E = 119° -(- 15°.6) =134°.6.
To check this, we can now solve the same problem in the
inverse way with the first deviation table. For the compass
course 134°.6, this table gives the deviation as — 5°.9. The
variation being — 10°, we have :
E = V + D = -10° - 5°.9 = - 15°.9 and
T = C + E = 134°.6 - 15°.9 = 118°.7,
agreeing very closely with the true course 119°, with which
we started. This shows that the two deviation tables are
quite consistent in this case, and also checks the accuracy
of the calculation.
52
NAVIGATION
We shall close this chapter with the following little table,
showing the correspondence between the two methods of
dividing the compass card into points, and into degrees.
COMPASS POINTS AND DEGREES
- ,
o ,
o ,
o ,
North
0 0
East
90 0
South
180 0
West
270 0
N. by E.
11 15
E. by S.
101 15
S. by W.
191 15
W. by N.
281 15
N.N.E.
22 30
E.S.E.
112 30
s.s.w.
202 30
W.N.W.
29230
N.E. by N.
33 45
S.E. by E.
12345
S.W. by S.
21345
N.W. byW.
303 45
N.E.
45 0
S.E.
135 0
S.W.
225 0
N.W.
315 0
N.E. by E.
56 15
S.E. by S.
146 15
S.WibyW.
236 15
N.W. by N.
326 15
E.N.E.
6730
S.S.E.
157 30
W.S.W.
24730
N.N.W.
33730
E. by N.
7845
S. by E.
16845
W. by S.
25845
N. by W.
34845
J pt. = 2° 49'
i pt. = 5° 38'
pt. = 8° 26'
1 pt. = 11° 15'
CHAPTER V
COASTWISE NAVIGATION
BEFORE proceeding to a consideration of navigation by
means of astronomic observations, as it is practiced on the
high seas, we must first explain certain methods by which
it is possible to ascertain a ship's position in latitude and
longitude while she is in sight of land. Often such methods
suffice to complete a long coastwise voyage in safety; they
are always important for a last determination of the ship's
position before a deep-sea voyage actually begins. Such a
last determination is called "taking a departure" (cf. p. 2),
and from such point of departure dead-reckoning calcula-
tions begin for the first day of the voyage.
Any determination or fixing of a ship's position, by astro-
nomic observations or otherwise, is often called, for brevity,
a "fix." To obtain one while in sight of land it is customary
to make observations upon well-known objects ashore,
such, for instance, as lighthouses, or other conspicuous
objects marked on the chart. It is always possible to ob-
serve the bearings of such objects from the ship's deck with
the compass, azimuth circle, or pelorus (p. 44).
When there is but one such object in sight, it is impossible
to secure a fix with ordinary instruments, if the vessel is
at anchor. But if she is running, it is merely necessary to
take two bearings, and to estimate. the distance run by the
ship in the interval between the two. Figure 9 will make
this matter clear. A lighthouse ashore is at L. SS" is the
direction of the ship's course; S her position when the
first bearing was observed, and S' her position at the time
of the second bearing. SN is .the direction of the north.
53
54 NAVIGATION
After taking the first bearing, the navigator must calculate
ythe angle S"SL, between the ship's course SS" and the
lighthouse direction SL. Thus,
if the ship's course angle NSS"
(p. 10) was 20°, and the bearing
NSL was 42°, the angle S"SL
would be 42° - 20° =22°. As
the ship proceeds on her course,
the angle S"SL will become
larger, and a second bearing must
be taken at the moment when
the ship reaches the point S',
where the angle S"SL has become
S"S'L. This point S' must be
so chosen that the angle S"S'L
is just twice the angle S"SL ob-
served at S ; or, in this case, 44°.
This is called " doubling the bear-
FIG. 9.— Ship's Position by Two ing from the bow/' and it can
easily be accomplished if we con-
tinue watching the compass bearing of L as the ship goes
ahead, and catch the observation at the right moment. The
ship's course not having been changed from 20° (this is
important), the right moment will occur when L bears
20° + 44° =64° by the compass.
It can easily be proved by geometry that the distance
*S'L between the ship at S' and the lighthouse at L will be
equal to the distance SS' traveled by the ship in the inter-
val between the two observations. This distance can be
estimated quite accurately with an instrument called a
"log," or "patent log," which is towed astern of the ship.
It is so constructed that it turns as it is pulled through the
water, and the number of turns is automatically counted by
an attached contrivance on deck. The count is (also auto-
matically) turned into miles of distance ; so that the log on
deck will indicate how far the ship traveled from S to S'.
COASTWISE NAVIGATION 55
As soon as we know the distance S'L and the bearing of
the line S'L, we can "lay down" or "plot" the position of
S' on the chart; and this will be a "good fix." To do this,
let us indicate by B' the bearing of the line .S'L, and then
draw on the chart, through the lighthouse L, a pencil line
whose bearing from L is B' + 180°, or "Bf reversed." This
can be done with a "course protractor," or with "parallel
rulers," instruments to be purchased from any dealer in
navigators' supplies. Next we measure or "lay off" on that
line the distance S'L, equal to the run SS' as it came from
the log. We always know the right "scale" of the chart
(or fraction of an inch corresponding to one logged mile)
which must be used in laying off the distance S'L; for we
know that one mile always corresponds to 1 minute of
latitude (p. 15), and the right- and left-hand edges of the
chart are always divided into degrees and minutes of latitude.
Since the above bearings were observed by compass, it
is now important to consider the compass error (p. 43).
This will not affect the observations, because it will be the
same for both ship's course and lighthouse bearing, so the
angles S"SL and S"S'L, which are obtained by subtraction,
will be the same as if there were no compass error. But
when we come to plotting on the chart, the compass bearing
B' must be corrected by adding the deviation from the
deviation table (pp. 48, 49). The resulting magnetic bear-
ing (p. 49) must be used for B', if the chart has printed
on it a compass card (p. 41) showing magnetic bearings.
If the printed card shows true bearings only, B' must be
corrected for both deviation and variation (p. 43).
A specially important case of the foregoing occurs when
the two angles S"SL and S"S'L are 45° and 90°. The
second bearing B' will then put the light just abeam, and
the distance by log, SS', is the distance, at which the ship
passes the light abeam. This case is called a "bow-and-
beam bearing." The navigator sights the light when it bears
45° or 4 points (p. 52) "broad" on the bow, "starboard,"
56 NAVIGATION
or "port." He then "reads" the log. When he brings
the light abeam through the motion of the ship, he reads
the log again, and the run in the interval, as taken from the
log, is the light's distance abeam.
When sailing along the coast, it is particularly important
so to shape the ship's course that lights and other promi-
nent landmarks will be passed at the right distance abeam.
The chart shows what the right distance is : if the navigator
shapes a course which makes the distance abeam too small,
he may fail to clear rocks or shoals extending seaward ; and
if he makes it too large, he may lengthen his voyage unneces-
sarily in rounding the light.
There are certain pairs of angles (S"SL and S"S'L) which
will make known the coming distance abeam long before
the ship is dangerously near the light. These angles, S"SL
and S"S'L, are called "bearings from the bow" (see p. 54),
since they are really measured from the ship's bow instead
of the north. If the two bearings from the bow are either
of the following pairs :
22° and 34°, 32° and 59°,
27° and 46°, 40° and 79°,
then the logged distance in the interval between the two
observations is the distance at which the ship will pass the
light abeam if she continues on her present course. This
kind of observation will inform the navigator whether his
course is safe in ample time to change it if necessary ; and,
since in this case no bearings are marked on the chart, no
attention need be paid to compass error.
When two or more known and conspicuous landmarks
are visible from the ship, it is possible to secure a fix by
means of "cross-bearings." Observe the bearings of the
objects as nearly simultaneously as possible. Allow for
compass error in the manner just explained. Calculate
for each object a reversed bearing by adding 180° to its
observed bearing. Draw on the chart through each object
COASTWISE NAVIGATION
57
a pencil line having the proper reversed bearing and these
lines will intersect at the point on the chart where the ship
is located. Figure 10
illustrates this matter.
L, L', L" are lights or
landmarks ashore,
visible from the ship,
and also printed on
the chart. The ship
is at S. The lines in-
tersecting at S repre-
sent the reversed
bearings of L, L', L",
as observed from S.
Only two lines are nec-
essary ; and they
should be chosen so
that the angle be-
tween them is as near
FIG. 10. — Ship's Position by Cross Bearings.
a right angle as possible, if high accuracy is required in the
fix. The third object and line merely serve as an additional
check or safeguard against error.
In addition to the foregoing methods of locating a ship
by observations of objects ashore, there is a way to avoid
sunken rocks or shoals without actually locating the ship
on the chart. It is called the "danger angle," and is shown
in Fig. 11. The small circle is supposed drawn on the chart
around a rocky shoal K which must be cleared by the ship
traveling along the course SSf. To make certain of clearing
it safely, the navigator selects two visible objects ashore,
and shown on the chart at L and L'. He draws on the
chart a large circle passing through L and L', and just touch-
ing the dangerous small circle at T. There is no difficulty
in finding the center of the large circle, because it must be
somewhere on the line PQ, which is drawn at right angles
to the line LL' at its middle point P. A few trials with a
58
NAVIGATION
pair of compasses will locate the center. Next, the two lines
LT and L'T are drawn. Then the angle LTL' is called the
danger angle.
Now it is a principle of geometry that if we select other
points on the large circle, such as T' and T", the angles
FIG. 11. — The Danger Angle.
LT'U, LT"L', etc., will all be equal, and will contain the
same number of degrees as the danger angle LTL'. It fol-
lows that if the navigator measures from the deck the angle
formed by two lines drawn to the ship from L and L', and
if he finds it equal to the danger angle LTL', as measured
on the chart with a protractor (p. 55), he then knows that
the ship is somewhere on the large circle, and is therefore
perhaps too near the small dangerous circle. If, on the
other hand, the ship is entirely outside the large circle, and
therefore surely safe from the gangers of the small circle,
COASTWISE NAVIGATION
59
the measured angle at the ship between the objects L and
L' will always be smaller than the danger angle LTL'.
Angles can be measured from the deck by taking compass
bearings of L and L'. The difference of the two will be the
deck angle, which should be smaller than the danger angle
measured on the chart. But the very best way to measure
the deck angle is to use the sextant, an angle-measuring
instrument to be described later (p. 61).
The danger angle can also be used when it is necessary to
pass between a sunken danger circle and the shore. The
large circle is then drawn through L and L' as before, but in
such a way as just to touch the inside of the small circle
instead of the outside. To pass inshore of the small circle
it is then necessary for the navigator
to keep his measured deck angle larger
than the danger angle, instead of
smaller.
Navigators also use at times a
means of safety known as the " danger
bearing," illustrated in Fig. 12.
There is but one charted object in
sight ashore at the point L. The ship
at S must steer in such a way as to
avoid sunken rocks at K. Evidently,
she must pass outside the line SQ, of
which the bearing from the north is
the angle NSQ, which can be meas-
ured on the chart. This is the danger
bearing, and the ship's course SS', to
be safe, must be greater than the danger bearing. In the
case shown in the figure, the danger bearing would be very
useful long before a fix could be had by means of bearings
from the bow or bow-and-beam bearings.
Finally, to complete this part of our subject, it is neces-
sary to mention "soundings," which are a method of feel-
ing the land, even when it cannot be seen. By means of
FIG. 12. — The Danger
Bearing.
60 NAVIGATION
the "lead-line" the mariner can ascertain when he is in
shoal water ; and as depths of water are always marked on
the chart, he can often get valuable information as to the
ship's position. As she runs along her course, he can take
a "line of soundings" and upon examining the chart he
will often find but a single possible line on the chart where
the charted depths correspond with those observed. It
follows that the ship's course must have been along that
line on the chart ; and at an anxious moment, in a fog, such
a check will be a great relief to the navigator. Even in
the ocean, far from land, it is possible to take soundings
with the "sounding machine" at great depths, and in some
parts of the ocean quite accurate locating of the ship will
result. Specimens from the ocean floor can also be brought
up by attaching some sticky grease to the bottom of the
lead, and at times these specimens also give information
of value, for the charts always specify the kind of bottom
existing in various parts of the ocean.
CHAPTER VI
THE SEXTANT
WE have twice made reference to this instrument — once
(p. 5) as a contrivance for ascertaining by observation how
high the sun is in the sky, and again (p. 59) in the measure-
ment of the danger angle. These two uses of the sextant
are not inconsistent, for it is really intended for the measure-
ment of any angle (p. 8) formed at the observer's eye by
two lines drawn to two distant objects. In the case of the
danger angle these two distant objects are landmarks
ashore; in the case of the sun they are the "horizon" line
(where sea and sky seem to meet), and the sun itself. This
height of the sun (or of any star) in the sky is called its
" altitude"; and so the altitude is always an angle, to be
measured in degrees and minutes. The point directly over-
head is the "zenith"; the angle between lines drawn to
horizon and zenith is 90°, or a right angle. An altitude of
40°, for instance, simply means that the distance from the
horizon to the sun is f$ of the total distance from horizon
to zenith.
Figure 13 will give an idea of the construction of the sex-
tant.1 The essential parts are two small silvered mirrors,
M and ra; a telescope, EK; and a circle, A A, engraved
with "graduations," by means of which angles may be
measured upon it in degrees, minutes, and seconds. The
mirror m and the telescope EK are firmly attached to the
sextant ; but the mirror M is pivoted in such a way that it
1 Quoted in part from Jacoby's " Astronomy, a Popular Hand-
book," Macmillan, 1913 ; reprinted 1915.
61
62
NAVIGATION
can be turned, and the angle through which it is turned
measured on the circle by means of the index CB. When
the mirror M is turned until it is parallel to the fixed mirror
m, the circle "reads" or indicates 0°, because the angle be-
tween the two mirrors is then 0°. In all other positions
FIG. 13. — The Sextant.
of the mirror M the circle measures the angle between the
two mirrors. P and Q are sets of colored glasses, which can
be interposed temporarily, when the sun's rays are so bril-
liant as to be hurtful to the observer's eye. R is a small
magnifying glass, pivoted at S, intended to facilitate the
examination of the index CB. At C and B are shown the
"clamp," by which the index can be fastened to the circle,
and the "tangent screw," or "slow-motion screw" which
will adjust it delicately, after it has been clamped. / and F
are additional telescopes or accessories.
The mirror m has an important peculiarity. The silver-
ing is scraped away at the back of the mirror from half its
THE SEXTANT 63
surface. Thus only one half reflects ; the other half is
simply transparent glass. A navigator looking into the
telescope at E will therefore look through the mirror m with
half his telescope, and with the other half he will look into
the mirror.
Now it is a fact that half a telescope acts just like a whole
one. If a person using an ordinary spy-glass half covers
the big end with his hand, he will see the same view he saw
with the whole glass. Only, as half the "light-gathering"
power is cut off, this view will be fainter, — less luminous.
Applying this to the sextant telescope, it is clear that the
observer will see two things at once : with half the telescope
he will see what is visible through the mirror m; and with
the other half he will see what is visible by reflection from
the mirror m.
If he holds the sextant in such a position that the telescope
is horizontal, while the frame of the instrument is vertical,
he will see the visible sea horizon with half the telescope
through the mirror m. If the other mirror M is then turned
to the proper position, it is possible to see the sun in the sky
at the same time, with the other half of the telescope, the
solar rays having been reflected successively from both mir-
rors, M and m. To make this possible, the sextant tele-
scope must be aimed at that point of the sea horizon which
is directly under the sun. The solar rays will then strike
the mirror M first ; be thence reflected to the silvered part
of the mirror m; and finally reflected a second time into
the telescope. Therefore the observation consists in so
turning the movable mirror M, that the sun and horizon
can be seen coincidently in the telescope.
The angle between the mirrors can then be measured
on the circle ; and it is easy to prove by geometry that the
angular altitude of the sun will be twice the angle between
the two mirrors. Thus it should merely be necessary to
double the mirror angle, as indicated by the sextant index,
to obtain the solar altitude. But the sextant makers always
64 NAVIGATION
save the navigator the trouble of doubling the angle by the
simple device of numbering half degrees on the arc AA as
if they were whole degrees ; so the angle as it comes from the
sextant is already doubled for further use. The mirror m
is called the "horizon glass," because the navigator looks
through it at the horizon. The other mirror M is the "index
glass," because it is attached to the index arm.
When the sextant is used for non-astronomical observa-
tions, such as the danger angle, the frame is held horizontally,
instead of vertically, as in observations of the sun. The
telescope is aimed at the left-hand object ashore, and that
object is viewed through the horizon glass m. The index
glass M is then turned until light from the right-hand object
is also brought into the telescope, after successive reflections
from the two mirrors M and m. The two objects will then
be seen "superposed," and the sextant arc will give the
angle between two lines drawn from the observer on board
to the two objects ashore. This angle should be smaller
than the danger angle to keep the ship safely off-shore of
sunken dangers (p. 59).
Reading the sextant circle, or ascertaining from it the
angle that has been measured, is accomplished by means of
a "vernier." This is a short circular arc, engraved with
graduations resembling those on the sextant circle, attached
to the index CB (fig. 13) just under the little magnifier R.
It is so placed that the graduations on the sextant circle
and the vernier are close together and can be seen at the
same time through the magnifier R. Figure 14 gives an idea
of the vernier and a part of the sextant circle near the zero
of its graduations. Numbers on both circle and vernier
increase toward the left. On the circle, the largest spaces,
marked by long lines, are whole degree spaces. Each is
usually divided into two halves of 30' each indicated by
shorter lines, and these are again subdivided into three
small spaces of 10' each. The divisions on the vernier
resemble those on the circle, except that the degree spaces
THE SEXTANT
65
of the former are here called min-
ute spaces, and the 10' spaces of
the former are called 10" spaces.
The real index of the instru-
ment is the zero mark on the
vernier, sometimes provided with
an engraved "arrow." If this
falls exactly on a degree mark of
the circle, say the 1° mark, the
reading of the instrument is ex-
actly 1° 0' 0". If it falls exactly
on a small line of the circle, say
the second to the left of the 1°
mark, the reading is exactly 1°
20' 0". But if it falls between two
of the small lines, say between
the 20' and 30' marks to the left
of the 1° mark (as shown in the
figure), the reading must be 1°
20' and a "bit." It is the busi-
ness of the vernier to estimate
the size of that bit. To do this
look along the vernier until you
find a line which is exactly op-
posite some line on the circle.
There will always be such a line :
in the figure it is the 6' line of the
vernier. Pay no further atten-
tion to noting which line on the
circle is the one thus " exactly
opposite"; it matters not which
line it is. But read carefully the
number on the vernier belonging
to the "exactly opposite" line
you have found there. Being on
this occasion the 6' line, it follows
66 NAVIGATION
that the bit is 6' ; and as we found the reading to be 1° 20'
and a bit, the complete reading is 1° 20' + 6' = 1° 26'.
If the vernier line that happened to be "exactly opposite"
was not one of the ten long minute lines, but fell between
two of them, it would indicate that the bit was made up of
minutes and seconds, instead of being an exact number of
minutes. For each space the "exactly opposite" vernier
line happens to lie to the left of a long vernier minute line,
10" must be added to the bit. For instance, if in the figure
the "exactly opposite" vernier line was the next short one
to the left of the 6' long line, the bit would be 6' 10", and the
complete reading 1° 26' 10", instead of 1° 26'. But seconds
are not really required when observing aboard ship, so that
it will be sufficient, in using the vernier, to find the number
of the long vernier line that comes nearest to being "exactly
opposite."
It will also be noticed in the figure that the sextant circle
has some additional graduations to the right of the 0° mark.
These are called "off the arc" graduations, and it is some-
times necessary to read a small angle upon them, measuring
from the 0° mark to the right instead of the left. This makes
it necessary to read the vernier backwards, calling the 0'
mark of the vernier 10' and the 10' mark 0'.
This backward reading of the vernier offers no particular
difficulty, and it is especially useful in determining by ob-
servation the "index error" of the sextant. We have seen
(p. 62) that when the two sextant mirrors are parallel,
the index should read 0° 0' 0". But it is seldom possible
to adjust the instrument so that this condition will be satis-
fied exactly ; nor would the adjustment remain perfect very
long. A better plan is to determine by observation how
much the reading differs from 0° 0' 0", when the mirrors
are parallel. This difference is the index error, and must
be applied as a correction to all angles observed with the
instrument.
It is easy to make the mirrors parallel : we have merely
THE SEXTANT 67
to sight some distant well-defined terrestrial object like the
gilt ball on the top of a flagpole (or the sea horizon, if aboard
ship at sea), after clamping the index near 0°. We shall
then see in the telescope two images of the distant object;
one by direct vision through the unsilvered part of the hori-
zon glass, the other after reflection from both mirrors. By
means of the tangent screw, the observer, with his eye at
the telescope, can bring these two images together, so that
they will appear as a single image. Then the mirrors will
be parallel, and the vernier should read 0° 0' 0". If it actually
reads 0° 8', for instance, instead of 0° 0' 0", it means that the
reading is 8' too large on account of index error ; and every
angle measured with that sextant at that time will be 8'
too large, and must be corrected by subtracting 8' from it.
If, on the other hand, the reading is 8' "off the arc,"
when it should be 0° 0', the instrument reads 8' too small,
and any angle measured with it must be corrected by adding
8' to it.
For accurate determination of the index error (and it
should be checked frequently), navigators prefer to observe
the sun, or at night, a star. If a star is used, the process
is the same as just described for a flagpole ball. But if
the sun is used, a slightly different method is required. The
sun, as seen in the telescope, shows a round disk of con-
siderable size, and it is not possible to
superpose the two images accurately.
Therefore it is better to make them
just touch, as shown in Fig. 15, when
they are said to be "tangent" to each
other. This must be done successively
in two positions, AB and BA. In
,, , ., ,, f, , .,, ,, FIG. 15. — Index Error.
other words, after the first "tangency
has been observed, the tangent screw (B, fig. 13) is manipu-
lated until the image A passes across B from top to bottom,
and gives a new tangency in the second position.
Each tangency will give a reading of the vernier. Unless
68 NAVIGATION
the sextant is greatly out of -adjustment, one of these read-
ings will be off the arc, the other on the arc. If there were
no index error, the off-arc and on-arc readings would be
equal; if they differ, half the difference is the index error.
If the off-arc reading is the larger, all altitudes measured
with that sextant must be increased by the amount of the
index error ; and if the on-arc reading is the larger, all such
altitudes must be similarly diminished.
The following is an example of an index error determina-
tion:
On-arc readings, Off-arc readings,
31' 20" 33' 20"
31 40 33 50
30 50 34 0
Means, 31' 17" 33' 43"
The difference is 33' 43" - 31' 17" = 2' 26". Half the
difference, or 1' 13", is the index error ; and because readings
on the arc are the smaller, all angles read with this instru-
ment must be increased by 1' 13", or, for ordinary purposes
of navigation, by 1'.
In addition to certain "adjusting screws" with which
the index error can be reduced when it becomes unduly
large, means are provided for three other sextant adjust-
ments. These are :
1. To make the index glass perpendicular to the frame of
the instrument.
2. To do the same with the horizon glass.
3. To set the telescope parallel to the frame of the instru-
ment.
These adjustments are always completed by the maker
before a sextant is sent out, nor does the navigator usually
need to correct them himself. But it is important to know
how to test them occasionally. Perpendicularity of the
index glass can be examined by looking into the glass very
obliquely with the index set near 0°. It is then possible to
see the inner edge of the sextant circle both by looking at
THE SEXTANT 69
it directly, past the edge of the index glass, and also by reflec-
tion in the glass itself. The inner edge of the circle should
form a continuous line when so examined, if the glass is
perpendicular ; but if it is inclined, the line will appear broken,
instead of continuous.
Secondly, perpendicularity of the horizon glass can be
tested at the same time the index error is determined by
observing a star or a distant terrestrial point (p. 67). The
index glass having been properly adjusted to perpendic-
ularity, the two mirrors can never be made parallel by
moving the index, unless the horizon glass is also properly
perpendicular. Any existing lack of adjustment will there-
fore betray itself in the index error determination, because
the two images of the star or distant object will not be super-
posed in any position of the index.
Thirdly, the parallelism of the telescope to the frame of
the instrument can usually be best tested with an ordinary
pair of "calipers."
Having thus described the sextant, its adjustments, and its
use from the deck, we have still to explain how it can be used
ashore. Sometimes it is necessary for the 5,
navigator to make observations ashore,
when it is not usually possible to see the
horizon line (p. 61). Recourse must
then be had to an "artificial horizon,"
which is simply an iron basin full of
mercury covered with a glass roof. The
mercury furnishes an almost perfectly
horizontal mirror, and the glass roof
prevents wind from ruffling the mercury
surface, and thus destroying the mirror. pIG> 15. _ Artificial
Figure 16 explains the principle of the Horizon,
artificial horizon. HH is the mercury mirror, S the sun,
and X the sextant. The observer aims the sextant telescope
at the mercury where he can see a reflection of the sun. He
then measures with the instrument the angle between a line
70
NAVIGATION
drawn to the sun as seen reflected in the mercury and another
line drawn to the actual sun in the sky. It can be shown
by geometry that this measured angle will be just twice the
real altitude of the sun, such as it would be if observed from
the sea horizon. Therefore, in using the artificial horizon,
it is merely necessary to divide the sextant angle by 2 to ob-
tain the correct altitude of the sun.
In observations of this kind two "suns" are seen at the
same time in the telescope, just as is the case in index error
observations (p. 67) ; whereas in observing from the sea
horizon, the telescope shows only one solar image and the
horizon line. When there are thus two solar images, they
must be brought into tangency, just as we have already
explained for index error (p. 67). When there is but one,
it must be brought into tangency with the visible sea
horizon line.
But this altitude is not yet ready to be used in the further
calculations for obtaining the position of the ship in latitude
and longitude. Further pre-
paratory corrections must be
applied, in addition to the
index error (p. 66), which is
always the first correction to
receive attention . These pre-
paratory corrections are :
1. "Dip" of the sea hori-
zon, due to the elevation of
the navigator on the ship's
deck above the surface of the
sea. Its cause is shown in
Fig. 1 7 . C is the center of the
Fio. 17.— Dip of the Horizon. earth, K a point at sea level,
and 0 the navigator, elevated
a distance OK above the sea. OZ is the direction of the ze-
nith (p. 61), OS the direction of the sun, and OH a horizontal
line from 0, OT is a line drawn through 0, and just touch-
THE SEXTANT 71
ing the sea surface at T'. Evidently OT will be the direc-
tion of the sea horizon, where sky and sea seem to meet.
Therefore, the altitude of the sun, as measured from the
visible sea horizon, will be the angle SOT ; whereas the angle
we require is the angle SOH, or the altitude of the sun
above the true horizontal line OH. Therefore the angle
HOT is a correction for dip which must be subtracted from
all measured altitudes, and the amount of the correction
depends on the height of the navigator's eye above the sea
surface.
2. "Refraction" is a bending of the light rays as they
come down to us from the sun through the terrestrial atmos-
phere. It always makes the sun seem higher in the sky
than it really is, giving another subtractive correction for
the observed altitude. The bending here involved is due
to the passage of the sun's light rays through atmospheric
strata of increasing density as the light approaches the
earth's surface.
3. "Parallax" is a small correction which must be added
to the observed altitude of the sun. In strict theory, all astro-
nomic observations are supposed to be made from the earth's
center instead of its surface where the ship floats; and the
small parallax correction allows for this minor theoretic
point. In the case of star observations this correction is
zero.
4. " Semidiameter " is a correction depending on the
choice by the navigator of a particular point on the sun's
disk (p. 67) for observation. The sun's altitude, as used
in the further calculations, should be the altitude of the sun's
center ; but it is impossible to locate the center of the disk
accurately in the telescope, so the navigator always observes
the lowest point of the disk. This is called the "lower
limb" of the sun.
Beginners sometimes have difficulty in distinguishing
the upper from the lower limb in the telescope. The best
way to do this is to focus the telescope on some distant
72 NAVIGATION
object, and note whether it appears upside-down in the
field of view. If so, the telescope is an "inverting" one,
and the top of the sun must be observed, as it appears in
the telescope, though it will really be the correct (or lower)
limb, because of inversion by the telescope. When using the
artificial horizon with an inverting telescope, the tangency
must be made by bringing the bottom of the mercury image
in contact with the top of the other image. The high-pow *
ered telescopes supplied with good sextants are usually in-
verting telescopes.
Evidently the measured altitude, as it comes from the
sextant, must be increased by the amount by which the sun's
center is higher than the lower limb, and this is the sun's
semidiameter. The index correction, together with the
above four additional corrections, will fully prepare a meas-
ured sextant altitude of the sun for further use in naviga-
tional calculations. In the case of a star, which appears
in the telescope as a point of light only, without any per^
ceptible disk, no semidiameter or parallax corrections are
required; and in using the artificial horizon (p. 69), no
correction for dip is necessary, either for the sun or a star.
It is possible to arrange these various corrections in con-
venient tables. Thus, in Table 6 (p. 247), we give a combi-
nation of corrections 2 (refraction), 3 (parallax), and 4 (semi-
diameter), to be used for observations of the sun's lower
limb, and the same combination without the semidiameter
and parallax l to be used for star observations. It will be
noticed that the tabular corrections vary for different values
of the observed altitude, which appears in the left-hand col-
umn of the table. This variation comes mainly from the
refraction part of the combined correction, for the refrac-
tion is much greater when the sun or star is observed at a
low altitude near the horizon than it is at a high altitude
near the zenith. At the foot of the page is given a small
supplementary correction depending on the date in the year.
1 Which leaves refraction only.
THE SEXTANT 73
This small correction is not important in navigation, but is
given here for the sake of completeness. It arises from the
semidiameter part of the combined correction, for the an-
nual orbit of the earth around the sun is of such a shape
that the earth is nearer the sun in January than it is in July,
which makes the sun appear bigger in January. And when the
sun appears big, the semidiameter will of course be large too.
Table 7 gives the dip of the sea horizon, the number in the
left-hand column being the height (in feet) of the navigator's
eye above sea level. This will be the height of the ship's
deck, increased by the height of the man's eye above the
deck. Unfortunately, the dip, as given in Table 7, at times
varies considerably from the dip as it actually exists at the
ship. The cause can be seen from Fig. 17 (p. 70), where
it will be noticed that the line from the observer at 0 to the
sea horizon at T' passes very near the surface of the ocean.
It is therefore entirely in the lowest strata of the terrestrial
atmosphere, and there quite irregular refractions sometimes
occur. These have been known to produce errors in the dip
amounting to 10' or 20', and it is principally the existence
of these unavoidable errors that makes it unnecessary to
read the sextant closer than the nearest minute (p. 66),
when observing from the deck. But when observing ashore
with the artificial horizon, which has no dip, the navigator
may, if he chooses, read seconds, especially if he intends to
use in his further calculations the "mean" or average of
a considerable number of observations.
We shall now give an example of the complete correction
of a sextant observation. Suppose the angle read from
the sextant was 30° 28', the index error (p. 68) 1', addi-
tive, height of observer's eye 26 feet. We should then
have :
observed altitude, lower limb = 30° 28'
index correction = + 1'
correction from Table 6 (p. 247) = + 14'
correction from Table 7 (p. 247) = - 5'
corrected altitude, for further use = 30° 38'
74 NAVIGATION.
If the altitude had been observed ashore with an arti-
ficial horizon, it might have been desirable to retain seconds.
The calculation might then have been as follows :
observed double altitude (see p. 70), lower limb = 63° 0' 20"
index correction (p. 68) = + 1 13
corrected double altitude =63 1 33
resulting altitude = 31 30 46
correction from Table 6 (interpolated) = + 14 31
corrected altitude, for further use =31 45 17
CHAPTER VII
THE NAUTICAL ALMANAC
BEFORE beginning the further utilization of altitude ob-
servations in our navigation calculations, it is necessary to
understand the use of the Nautical Almanac. This is an
annual publication, issued in two different editions by the
Nautical Almanac Office, United States Naval Observatory.
Copies can be obtained from the Superintendent of Docu-
ments, Washington, D. C., or through any dealer in nautical
supplies. Navigators do not need the larger edition, of which
the title is "American Ephemeris and Nautical Almanac";
accordingly, all our references are made to the smaller edi-
tion for the year 1917. Parts of certain pages from that
edition are reprinted in the present volume for convenience
of reference, and we shall give a somewhat detailed explana-
tion of the almanac page 29 (our p. 76).
Let us consider the date Monday, Dec. 17. We find for
that date, and for every even hour (0*, 2*, 4*, 6*, etc.) of
"Greenwich Mean Time" (abbreviated G. M. T.1), two
tabular numbers (p. 10) called "sun's declination" and
"equation of time."
To understand these it is necessary to bear in mind that
the kind of time in ordinary use is "solar time," as kept by
the sun. The "solar day" begins at "noon," called 0* in
astronomic navigation, and it continues through twenty-four
hours, without any confusing A.M. and P.M. In ordinary
life the day begins twelve hours sooner, at midnight, and
runs through two twelve-hour periods of A.M. and P.M. to
1 The reader is requested to note carefully this abbreviation, as
it will be used very frequently.
75
76
NAVIGATION
SUN, DECEMBER, 1917. From Nautical Almanac, p. 29
G. M. T.
SUN'S DEC-
LINATION
EQUATION
OF TIME
SUN'S DEC-
LINATION
EQUATION
OP TIME
SUN'S DEC-
LINATION
EQUATION
OP TIME
Monday 17
Tuesday 25
Saturday 29
h
0 /
m s
0 1
m s
0 /
m s
0
- 23 21.3
+ 3 56.8
- 23 24.7
-0 1.6
- 23 15.2
- 1 59.7
2
23 21.5
3 54.4
23 24.6
0 4.1
23 14.9
2 2.1
4
23 21.7
3 51.9
23 24.5
0 6.5
23 14.6
2 4.6
6
23 21.9
3 49.5
23 24.4
0 9.0
23 14.3
2 7.0
8
23 22.1
3 47.0
23 24.2
0 11.5
23 14.0
2 9.4
10
23 22.2
3 44.5
23 24.1
0 14.0
23 13.7
2 11.9
12
23 22.4
3 42.1
23 24.0
0 16.5
23 13.4
2 14.3
14
23 22.6
3 39.6
23 23.8
0 18.9
23 13.1
2 16.7
16
23 22.8
3 37.1
23 23.7
0 21.4
23 12.8
2 19.1
18
23 22.9
3 34.7
23 23.5
0 23.9
23 12.5
2 21.5
20
23 23.1
3 32.2
23 23.4
0 26.4
23 12.2
2 24.0
22
23 23.2
3 29.8
23 23.2
0 28.8
23 11.9
2 26.4
H. D.
0.1
1.2
0.1
1.2
0.1
1.2
Tuesday 18
Wednesday 26
Sunday 30
0
- 23 23.4
+ 3 27.3
- 23 23.1
- 0 31.3
- 23 11.6
- 2 28.8
2
23 23.6
3 24.8
23 22.9
0 33.8
23 11.3
2 31.2
4
23 23.7
3 22.3
23 22.7
0 36.3
23 11.0
2 33.6
6
23 23.8
3 19.9
23 22.5
0 38.7
23 10.6
2 36.0
8
23 24.0
3 17.4
23 22.4
0 41.2
23 10.3
2 38.4
10
23 24.1
3 14.9
23 22.2
0 43.7
23 10.0
2 40.9
12
23 24.3
3 12.5
23 22.0
0 46.2
23 9.7
2 43.3
14
23 24.4
3 10.0
23 21.8
0 48.6
23 9.3
2 45.7
16
23 24.5
3 7.5
23 21.7
0 51.1
23 9.0
2 48.1
18
23 24.6
3 5.0
23 21.5
0 53.6
23 8.6
2 50.5
20
23 24.8
3 2.6
23 21.3
0 56.0
23 8.3
2 52.9
22
23 24.9
3 0.1
23 21.1
0 58.5
23 7.9
2 55.3
H. D.
0.1
1.2
0.1
1.2
0.2
1.2
Wednesday 19
Thursday 27
Monday 31
0
- 23 25.0
+ 2 57.6
- 23 20.9
- 1 0.9
- 23 7.6
- 2 57.7
2
23 25.1
2 55.1
23 20.7
1 3.4
23 7.2
3 0.1
4
23 25.2
2 52.6
23 20.5
1 5.9
23 6.9
3 2.4
6
23 25.3
2 50.2
23 20.3
1 8.3
23 6.5
3 4.8
8
23 25.4
2 47.7
23 20.1
1 10.8
23 6.1
3 7.2
10
23 25.5
2 45.2
23 19.8
1 13.2
23 5.8
3 9.6
12
23 25.6
2 42.7
23 19.6
1 15.7
23 5.4
3 12.0
14
23 25.7
2 40.2
23 19.4
1 18.1
23 5.0
3 14.4
16
23 25.8
2 37.8
23 19.2
1 20.6
23 4.6
3 16.7
18
23 25.9
2 35.3
23 19.0
1 23.1
23 4.3
3 19.1
20
23 26.0
2 32.8
23 18.7
1 25.5
23 3.9
3 21.5
22
23 26.1
2 30.3
23 18.5
1 28.0
- 23 3.5
- 3 23.9
H. D.
0.0
1.2
0.1
1.2
0.2
1.2
Thursday 20
Friday 28
0
- 23 26.1
+ 2 27.8
- 23 18.3
^ 1 30.4
2
23 26.2
2 25.3
23 18.0
1 32.9
4
23 26.3
2 22.8
23 17.8
1 35.3
6
23 26.3
2 20.4
23 17.5
1 37.8
8
23 26.4
2 17.9
23 17.3
1 40.2
SEMIDIAMETER
10
23 26.5
2 15.4
23 17.0
1 42.6
12
23 26.5
2 12.9
23 16.8
1 45.1
14
23 26.6
2 10.4
23 16.5
1 47.5
Dec. 1
16'26
16
23 26.6
2 7.9
23 16.3
1 50.0
11
16'28
18
23 26.7
2 5.4
23 16.0
1 52.4
21
16'29
20
23 26.7
2 2.9
23 15.7
1 54.8
31
16'30
22
- 23 26.8
+ 2 0.4
- 23 15.4
- 1 57.3
H. D.
0.0
1.2
0.1
1.2
NOTE. — The Equation of Time is to be applied to the G. M. T. in accordance with
the sign as given.
THE NAUTICAL ALMANAC 77
the following midnight; but this "civil day," as it is called,
does not for the moment concern us.
Solar time, as kept by the visible sun, is a very incon-
venient kind of time, because there are certain peculiarities
in the astronomic motion of the earth which make these
solar days of unequal length. They are called "apparent
solar days" and the corresponding kind of time is "apparent
solar time."
To avoid the above inconvenience, an imaginary "mean
sun" and a "mean solar day" have been invented. The
mean sun conforms as nearly as possible to the average per-
formance of the visible sun, and the length of the mean
solar day is the average of all the apparent solar days through-
out the year. The corresponding kind of time, kept by the
mean sun, is "mean solar time" ; and this is the kind of time
recorded by all our watches and marine chronometers (p. 6).
The difference between these two kinds of solar time varies
on different dates, and even at different hours on the same
date. It is this difference which is called the "equation of
time " and which is one of the tabular numbers in the nautical
almanac page 29 (our p. 76).
This equation of time is of great importance in navigation,
and it is easy to see how page 29 of the almanac may be used
to find it. Suppose, for instance, we wish to know what the
equation is on Dec. 17, 1917, on board ship, when the ship's
chronometer indicates on its face 3 P.M., civil time, or (which
is the same thing) 3*, astronomical time (p. 75). Ship's
chronometers are always set to Greenwich mean time, so
that 3A by the chronometer signifies that the time at Green-
wich was 3\
We then look in the almanac page 29 (our p. 76), and find
that the equation was + 3W 54*.4 at 2h, G. M. T., and
+ 3m 5P.9 at 4*, G. M. T. Its value at 3* must be half-
way between these two, or + 3m 53*. 15. This we would
call + 3m 53*.2, so as to avoid the use of hundredths of
seconds, which do not need attention in navigation. And
78 NAVIGATION
since the equation is merely the difference between the
two kinds of solar time, the + sign means that it must be
added to G. M. T., to obtain Greenwich apparent time, in
accordance with the "Note" at the foot of the almanac
page 29. Consequently, the G. M. T. by chronometer having
been 3h Om 0*, the Greenwich apparent time at the same in-
stant was 3* 0"1 0» + 3m 53f.2 = 3* 3OT 53*.2.
It will be noticed that the process we have here used for
obtaining the equation from the almanac is merely an inter-
polation (see p. 12). Let us, as another example, find the
equation for Sunday, Dec. 30, at 10* 26m A.M., civil time by
chronometer, and we have purposely here retained the
civil method of reckoning time to make certain that the
reader understands the difference between civil and astro-
nomic (or navigation) time. The given time is 10* 26m A.M.,
civil time, Dec. 30. But the astronomic Dec. 30 does not
begin until noon (p. 75), so that it is not yet Dec. 30 by
astronomic reckoning. By that reckoning it is really only
22h 2Qm on Dec. 29. In other words, when the civil time is
P.M., as in the first example, the astronomic time is the same
as the civil time. But when the civil time is A.M., as in the
present example, the astronomic time is found by adding
12* to the civil time, and deducting 1 from the date. These
complications emphasize the advantage of the astronomic
count, which avoids A.M. and P.M. altogether.
We now have from the almanac (p. 76) :
equation of time, Dec. 29, 22A, G. M. T. = - 2m 2Q'A,
equation of time, Dec. 30, 0A, G. M. T. = - 2m 28'.8 ;
and the numbers in this example have been purposely so
chosen that the above two tabular values of the equation
(between which the required value falls) come from different
dates in the almanac. This creates no confusion, for these
two values of the equation are really consecutive tabular
numbers, just as much as if they occurred on a single date.
The difference between the two values of the equation is
THE NAUTICAL ALMANAC 79
2*.4; and as this difference corresponds to 2h in the left-
hand (or argument) column, it follows that the difference
for lh is here P.2. This is the change of the equation per
hour of time; it is called the "hourly difference" (abbre-
viated H. D.) and is printed in the almanac at the foot of
each daily column.
Now we want the equation for Dec. 29, 22A 26TO, by the
chronometer. The 26™ must next be changed into a decimal
fraction of an hour. 26 m = ff of an hour = 0A.43. So the
time for which we want the equation becomes Dec. 29,
22A.43. The H. D. being P.2, the change in OM3 will be
1'.2 X 0.43 = 0*.5. The almanac shows that at 22A the equa-
tion was 2OT 26*. 4, and was increasing numerically. There-
fore, at 22A.43, it was 2m 268.4 + 0'.5 = 2m 26'.9. And this
number has the minus sign. Therefore, the G. M. T. being
Dec. 29, 22* 26m, the Greenwich apparent time at the same
instant will be Dec. 29, 22* 26m - 2m 26*.9 = Dec. 29,
22* 23™ 33M.
Most of these minor interpolation calculations, which are
here set forth in great detail for the benefit of the beginner,
can be made with sufficient accuracy by a skilled navigator
mentally.
In the foregoing two examples we have assumed that the
chronometer was right, but these instruments practically
never run quite correctly. Therefore, before leaving port,
navigators always have their chronometers "rated" by a
chronometer expert; and when the instrument is returned
to the ship just before sailing, a "rate card" (or "rate paper")
always comes with it. Let us suppose that in the present
example this card stated that the chronometer was slow
8m 22'. 5 x on Dec. 20, at noon, and was "losing" 2 18.8 daily.
The 8ro 22*. 5 would then be the "chronometer error" on
Dec. 20 ; and the 1*.8 would be its "daily rate."
1 This number is here purposely chosen much larger than would
ever occur in practice.
2 The opposite kind of "rate" is called "gaining."
80 NAVIGATION .,
From Dec. 20, noon, to Dec. 30, 10* 26TO A.M. is an interval
of 9 days 22 hours 26 minutes. This interval must now be
reduced to a decimal of a day. 26m = £$ of an hour = 0A.43.
The interval is therefore 9* 22A.43.
But 22A.43 =2-$£* days = Otf.93. Therefore, in days, the
interval is 9a.93. This transformation of hours and minutes
into decimals of a day can be accomplished with less trouble by
means of our Table 8 (p. 248).
Having a losing rate of P.8 daily, the chronometer lost
1'.8 X 9.93 = 17*.9 in the interval of 9.93 days. And as it was
already slow 8m 22s. 5 on Dec. 20, it was slow 8m 22*.5 + 17*.9
= 8m 40s. 4 at the time for which the equation is. required.
Now the equation was required for Dec. 29, 22* 26OT by the
chronometer; and that instrument being slow 8TO 40*.4, the
correct G. M. T. was : Dec. 29, 22h 26m + 8m 40*.4 = Dec. 29,
22* 34™ 40* .4. Turned into a decimal fraction of an hour,
this becomes Dec. 29, 22A.58, instead of 22hA3, as we found
before, when the chronometer error was omitted from the
calculation. The H. D. is 1*.2, as before, and the change
in ' 0A.58 = K2 X 0.58 = 08.7. Therefore, at 22A.58 the
equation is 2m 268.4 + 0*.7 = 2m 27M. This still has the
minus sign, so that the correct Greenwich apparent time
becomes Dec. 29, 22* 34"1 40'.4 - 2m 27M = 22A 32m 13S.3.
All the above calculations have been carried out here with
unnecessary accuracy. There would be no harm if the result
were in error by a few tenths of a second ; and it is this cir-
cumstance that makes it possible to perform these inter-
polations largely mentally.
In the foregoing examples no account was taken of the
ship's location on the ocean; yet this location may have an
indirect influence on the calculations. To understand this,
we must consider for a moment the time-differences which
exist between different places on the earth. The sun rises in
the east and travels across the sky toward the west ; so that
if we consider two places like Greenwich, England, and New
York, for instance, the sun, because of this motion from east
THE NAUTICAL ALMANAC 81
to west, will pass Greenwich first. Consequently, when it is
noon in New York, it has already been noon in Greenwich,
and is afternoon there. Greenwich time is therefore always
later than New York time. The same is true of any other
two places ; there is always a time-difference between them,
and the easterly place has the later or "faster" time.
The amount of such time-difference of course depends
on the relative location of the two places, and the relation is
such that 15° of longitude-difference corresponds exactly
to lh of time-difference. Thus Sandy Hook, which is in
longitude 73° 50' west of Greenwich, has a time-difference
from Greenwich of 4* 55 m 20*. This conversion of longitude
into time-difference is best accomplished by means of our
Table 9 (p. 249). According to that table :
73° = 4* 52* 0»
50' 3 20
73° 50'' = 4* 55"» 20»
The indirect influence of such time-differences upon the
use of the almanac is that they may at times, especially
when they are large, make the Greenwich date of the ob-
servation different from the date on board. Thus a vessel
off Manila Bay, in longitude 120° east of Greenwich, would
have her local time 8ft (120°) later than Greenwich time. If
a sextant observation was made on board at 4 P.M., civil
time, on a Thursday, the chronometer would indicate Sh,
and it would be 8 A.M. on Thursday, because Greenwich is
8h earlier than the ship. This 8 A.M. would really be 20* of
the preceding Wednesday by astronomic time, and so the
almanac date used would be one day earlier than the date
of the observation. The chronometer will always give the
right Greenwich time, but the navigator must be very care-
ful to interpolate the almanac numbers on the right date.
We have now learned how to ascertain the equation of
time from the almanac, and how to use it for transforming
G. M. T. into Greenwich apparent time. The contrary
transformation, from Greenwich apparent time to G- M. T.,
82 NAVIGATION
can be made by applying the equation in the opposite way :
subtracting when it has the + sign in the almanac, and add-
ing when it has the — sign.
The great importance of these time transformations comes
from the fact that sextant observations must necessarily be
made upon the visible sun. When they are made for the
purpose of calculating the local time on board, this local
time will therefore necessarily be local apparent solar time, as
kept by the visible sun. At the instant of the observation
(p. 6), the chronometer face (corrected for error and rate)
tells us the G. M. T. If this is turned into Greenwich ap-
parent time by applying the equation, we have only to com-
pare the Greenwich and the ship's apparent times to get
the time-difference between the ship and Greenwich. This
time-difference can then be turned into degrees and minutes,
and will be the ship's longitude. Examples of this calcu-
lation will be given in detail (p. 99). It is also worth
noting here that the time-difference between any two places
is precisely the same, quite irrespective of the kind of time
in which it is counted.
To complete our explanation of the almanac page 29 (our
p. 76), it remains to give an example of a calculation of the
sun's declination. This is an angle in degrees and minutes,
and it is interpolated just like the equation by the aid of
its H. D. Thus, for Dec. 29, 22*.58 (p. 80) the declination
is obtained thus :
Dec. 29, 22*, declination = 23° 1 1 '.9
H.D. (O'.l) x 0*.58 = 0.1, declination decreasing ;
by subtraction, at 22*.58, dec. = 23° 11 '.8,
and according to the almanac, this declination must be given
the minus sign. When the sign should be +, that fact is
indicated in the almanac. The use of the declination will
be explained later; the accuracy required in the interpo-
lation of it is not so great as we have used here, for the
nearest minute suffices in practically all navigation work.
In addition to the sun's declination, navigators require
THE NAUTICAL ALMANAC
83
in their further calculations another number called the sun's
"right ascension" (abbreviated, R. A.). This is obtained
from pages like the almanac page 3 (reprinted in part below).
It is always the R. A. of the "mean sun" that we need,
and the almanac gives it for Greenwich mean noon of each
day in the year. When needed in our further calcula-
tions, it is of course always required for the exact moment
when a sextant observation was made. In fact, this state-
ment applies also to the equation of time and declination.
They must always be interpolated from the almanac for the
moment when the navigator actually observed the sun ; and
SUN, 1917. From Nautical Almanac, p. 3
DAY
OF
MONTH
RIGHT ASCENSION OF THE MEAN SUN AT GREENWICH MEAN NOON
July
August
September
October
November
December
i m a
h m a
h m a
h m s
h m a
h m s
1
6 35 52.2
8 38 5.5
10 40 18.7
12 38 35.3
14 40 48.4
16 39 5.1
2
6 39 48.8
8 42 2.0
10 44 15.2
12 42 31.8
14 44 45.0
16 43 1.7
3
6 43 45.3
8 45 58.6
10 48 11.8
12 46 28.4
14 48 41.5
16 46 58.2
4
6 47 41.9
8 49 55.1
10 52 8.3
12 50 24.9
14 52 38.1
16 50 54.8
5
6 51 38.4
8 53 51.7
10 56 4.9
12 54 21.5
14 56 34.6
16 54 51.3
6
6 55 35.0
8 57 48.2
11 0 1.4
12 58 18.0
15 0 31.2
16 58 47.9
7
6 59 31.6
9 1 44.8
11 3 58.0
13 2 14.6
15 4 27.8
17 2 44.5
8
7 3 28.1
9 5 41.4
11 7 54.5
13 6 11.1
15 8 24.3
17 6 41.0
9
7 7 24.7
9 9 37.9
11 11 51.1
13 10 7.7
15 12 20.9
17 10 37.6
10
7 11 21.2
9 13 34.5
11 15 47.6
13 14 4.2
15 16 17.4
17 14 34.1
11
7 15 17.8
9 17 31.0
11 19 44.2
13 18 0.8
15 20 14.0
17 18 30.7
12
7 19 14.3
9 21 27.6
11 23 40.8
13 21 57.3
15 24 10.5
17 22 27.2
13
7 23 10.9
9 25 24.1
11 27 37.3
13 25 53.9
15 28 7.1
17 26 23.8
14
7 27 7.4
9 29 20.7
11 31 33.9
13 29 50.4
15 32 3.6
17 30 20.4
15
7 31 4.0
9 33 17.2
11 35 30.4
13 33 47.0
15 36 0.2
17 34 16.9
16
7 35 0.6
9 37 13.8
11 39 27.0
13 37 43.6
15 39 56.8
17 38 13.5
17
7 38 57.1
9 41 10.4
11 43 23.5
13 41 40.1
15 43 53.3
17 42 10.0
18
7 42 53.7
9 45 6.9
11 47 20.1
13 45 36.7
15 47 49.9
17 46 6.6
19
7 46 50.2
9 49 3.5
11 51 16.6
13 49 33.2
15 51 46.4
17 50 3.2
20
7 50 46.8
9 53 0.0
11 55 13.2
13 53 29.8
15 55 43.0
17 53 59.7
21
7 54 43.4
9 56 56.6
11 59 9.7
13 57 26.3
15 59 39.5
17 57 56.3
22
7 58 39.9
10 0 53.1
12 3 6.3
14 1 22.9
16 3 36.1
18 1 52.8
23
8 2 36.5
10 4 49.7
12 7 2.8
14 5 19.4
16 7 32.6
18 5 49.4
24
8 6 33.0
10 8 46.2
12 10 59.4
14 9 16.0
16 11 29.2
18 9 46.0
25
8 10 29.6
10 12 42.8
12 14 55.9
14 13 12.5
16 15 25.8
18 13 42.5
26
8 14 26.1
10 16 39.4
12 18 52.5
14 17 9.1
16 19 22.3
18 17 39.1
27
8 18 22.7
10 20 35.9
12 22 49.0
14 21 5.6
16 23 18.9
18 21 35.6
28
8 22 19.2
10 24 32.4
12 26 45.6
14 25 2.2
16 27 15.4
18 25 32.2
29
8 26 15.8
10 28 29.0
12 30 42.2
14 28 58.8
16 31 12.0
18 29 28.7
30
8 30 12.4
10 32 25.6
12 34 38.7
14 32 55.3
16 35 8.6
18 33 25.3
31
8 34 8.9
10 36 22.1
12 38 35.3
14 36 51.9
16 39 5.1
18 37 21.9
84
NAVIGATION
CORRECTION TO BE ADDED TO R. A. M. S. AT G. M. N. FOR
TIME PAST NOON
From Nautical Almanac, p. 3, Continued
TIME
Qtn
6"1
12-
18m
Mm
SO"1
36m
42m
«"»
TIME
h
12
13
14
15
m s
1 58.3
2 8.1
2 18.0
2 27.8
m a
1 59.3
2 9.1
2 19.0
2 28.8
m s
2 0.2
2 10.1
2 20.0
2 29.8
m s
2 1.2
2 11.1
2 20.9
2 30.8
m s
2 2.2
2 12.1
2 21.9
2 31.8
m s
2 3.2
2 13.1
2 22.9
2 32.8
m s
2 4.2
2 14.0
2 23.9
2 33.8
m s
2 5.2
2 15.0
2 24.9
2 34.7
m s
2 6.2
2 16.0
2 25.9
2 35.7
h
12
13
14
15
16
17
18
19
2 37.7
2 47.6
2 57.4
3 7.3
2 38.7
2 48.5
2 58.4
3 8.3
2 39.7
2 49.5
2 59.4
3 9.2
2 40.7
2 50.5
3 0.4
3 10.2
2 41.6
2 51.5
3 1.4
3 11.2
2 42.6
2 52.5
3 2.3
3 12.2
2 43.6
2 53.5
3 3.3
3 13.2
2 44.6
2 54.5
3 4.3
3 14.2
2 45.6
2 55.4
3 5.3
3 15.2
16
17
18
19
20
21
22
23
3 17.1
3 27.0
3 36.8
3 46.7
3 18.1
3 28.0
3 37.8
3 47.7
3 19.1
3 29.0
3 38.8
3 48.7
3 20.1
3 29.9
3 39.8
3 49.7
3 21.1
3 30.9
3 40.8
3 50.6
3 22.1
3 31.9
3 41.8
3 51.6
3 23.0
3 32.9
3 42.8
3 52.6
3 24.0
3 33.9
3 43.7
3 53.6
3 25.0
3 34.9
3 44.7
3 54.6
20
21
22
23
the Greenwich time of this event is of course always taken
from the chronometer (duly corrected for error and rate).
Thus, if the R. A. of the mean sun is required for Dec. 29,
22* 34m 40».4, G. M. T. (p. 80), we find from the almanac
page 3 (our p. 83) that the R. A. of the mean sun at Green-
wich mean noon is 18* 29m 28*.7.x This, according to the sup-
plementary table quoted above from page 3, must be increased
by a correction for "time past noon." In this case the time
past noon is 22* 34m 40*.4. The tabular correction for 22* 30™
is 3m 41'.8, and for 22* 36m it is 3m 42'.8. Ours falls between
these two, and an interpolation makes the correction 3m 42*.6.
Consequently, the R. A. of the mean sun for Dec. 29, 22*
34* 40».4, G. M. T. is 18* 29" 28'.7 + 3m 42'.6 = 18* 33m 11*.3.
It will be noticed that the small supplementary table
(quoted above from almanac page 3) only runs from 12* to 24*.
The other half of the table, from 0* to 12*, is printed on the
opposite page 2 of the almanac. There is also another
longer table, printed near the end of the almanac, and there
called Table III, from which the supplementary correction
can be taken without the necessity of interpolation.
It is not absolutely essential that the navigator learn what
1 Right ascensions are always thus measured in hours, minutes,
and seconds, like time, and they are counted from 0* to 24*.
THE NAUTICAL ALMANAC 85
the words "right ascension" and "declination" really mean.
But for the benefit of those who are curious in such matters
we may state that these numbers locate the position of the
sun (or of a star) on the sky. The sky is a great globe, called
by astronomers the "celestial sphere," and all heavenly
bodies are located upon it precisely as points on the earth
are there located by their latitudes and longitudes (p. 3).
There is a "celestial equator" with two "celestial poles,"
corresponding accurately to the terrestrial equator and poles.
Declination then corresponds exactly to latitude on the earth,
and so it measures the distance of a heavenly body from the
celestial equator. When the body is north of the celestial
equator, the declination is called +.
Right ascension similarly corresponds to longitude ; and for
the beginning point of right ascensions on the sky there is a
"celestial Greenwich," which is called the "vernal equinox."
After this brief digression into astronomy, we return to
our subject. We have seen (p. 82) that observations of
the sun will tell us only apparent solar time, because it is
only, the visible sun that we can observe. If the observations
are made upon a star, the kind of time is different from any
so far mentioned. It is called "sidereal time," or star time.
It is always possible to change mean solar time into sidereal
time, and vice versa, by a simple process of calculation ; but
the only change of this kind required in navigation is the
transformation of G. M. T. into Greenwich sidereal time.
To make this transformation, we have only to take from the
almanac, for the given G. M. T., the R. A. of the mean sun,
and then to add it to the given G. M. T.
Thus, to find the Greenwich sidereal time corresponding
to Dec. 29, 22* 34m 40*.4, G. M. T., we have already found
(p. 84) that the R. A. of the mean sun = 18* 33" 11*. 3
To this must be added the given G. M. T. = 22 34 40.4
Sum .= corresponding Greenwich sidereal time = 17*17m51*.7
1 The number of hours was here really 41* : but whenever it is
larger than 24*, we must drop or reject 24*.
CHAPTER VIII
OLDER NAVIGATION METHODS
WE shall now explain in detail certain standard methods
of determining a ship's latitude and longitude by means of
sextant observations. An understanding of these methods
is essential to a proper comprehension of the newer naviga-
tional processes to be described later ; and the older methods
are in fact still very widely used at sea, although most re-
cent authorities believe they should be rejected in favor of
the newer procedure.
The simplest of these older processes, and the one most
frequently employed, is the determination of the ship's
latitude by a noon or "meridian" observation ("noon-
sight") of the sun's altitude (p. 61). Now the sun is
higher in the sky at noon than it is at any other time during
the day ; and so it is possible to get the noon-sight by be-
ginning to observe the sun with the sextant a few minutes
before noon, and continuing the observation as long as the
sun's altitude is increasing. The moment it begins to
diminish, or the sun to "dip," as sailors say, the observation
should be terminated, and the vernier read.
The altitude thus observed will be an altitude of the lower
limb (p. 71) ; and before it is used further it must be fully
corrected for index error ; for refraction parallax and semi-
diameter ; and for dip ; all as in the example on p. 73,
where the observed altitude was 30° 28', and we found the
corrected altitude to be 30° 38'.
Next, the sun's declination must be taken from the al-
manac, being interpolated for the Greenwich time of the
86
87
observation, as in the example on p. 82, where we found
the declination to be - 23° 12' on Dec. 29, at 22* 34m 40'.4,
G. M.T. We shall suppose the above altitude 30*28' to
have been observed at the Greenwich time stated, so as to
make use of the results of our former calculated examples.
Nor is there any inconsistency in supposing a noon observa-
tion to have been made at 22* 34m 40*.4. For the noon
observation is made when it is noon on board ship, while
the 22* 34m 40v4 is the G. M. T. at the same moment.
The difference is simply the time-difference (p. 80) between
Greenwich and the ship.
The calculation of the ship's latitude is now made by the
following formula :
Latitude = 90° + Declination — Altitude.
In this formula, the plus sign signifies that the declination
must be added; and the minus sign signifies that the altitude
must be subtracted. Furthermore, it is most important to
remember that if the declination is itself a "minus declina-
tion," as in this example, the addition of it according to the
formula is really a subtraction. Or, in other words, and in
general, whenever a formula calls for an addition, and the
number to be added is a minus number, then that number
must be subtracted instead of added. And similarly, if the
formula calls for a subtraction, and the number to be sub-
tracted is a minus number, then that number must be added
instead of subtracted. Two minus signs neutralize each other.
In the present case we have, omitting seconds :
90° 0'
declination =-23 12
90° + declination = 66 48
altitude = 30 38
latitude = 36 10
In considering this result it is of interest to inquire where
this observation really locates the ship. Now we have not
yet stated what the date was, on board, when the observa-
88 NAVIGATION
tion was made ; but we have given the G. M. T. as Dec. 29,
22* 34m 40* .4. The noon-sight was taken, as a matter of
fact, afc noon on Dec. 30, or at the moment when the date
Dec. 30 commenced by astronomic reckoning. Therefore
the ship's time was later than the Greenwich time by about
1* 25" ; or 21° 15', allowing 15° to 1* (p. 81) ; and the ship
was (approximately) in 21° 15' east longitude from Greenwich.
This, together with the latitude 36° 10', locates the ship in
the Mediterranean, south of Greece, and west of Candia.
Although we have thus apparently located the ship com-
pletely in latitude and longitude from a single noon-sight,
it must not be supposed that we have really accomplished
this. The noon-sight is only suitable for ascertaining the
ship's latitude ; the longitude is determined so inaccurately
as to be practically useless. The reason for this is that
near noon the sun changes its altitude very slowly, because
it is then near the turning-point where its upward morning
motion is about to become a downward afternoon motion.
For the sun's daily motion in the sky is upward in the morn-
ing and downward in the afternoon. Near noon it runs
along horizontally, or very nearly so, for several minutes,
so that its altitude change is insignificant during that time.
It follows from this temporary invariability of altitude
that we cannot determine the exact moment when noon
occurs by observing altitude changes with the sextant. But
the latitude determination is not affected; because, for
the latitude, we only need to know the noon altitude. And
if we happen to measure it a little too soon or too late, on
account of the difficulty of fixing the moment of noon, no
harm will result, because the altitude very near noon is the
same as it is at noon precisely, 'as we have just seen.
It is, in general, practically impossible to determine both
latitude and longitude from a single observation. To deter-
mine two unknown things, at least two different observations
must be made. Nor can any skillful method of planning
the observation overcome this fundamental circumstance.
OLDER NAVIGATION METHODS 89
Returning now to our latitude formula (p. 87), it is
necessary to modify it somewhat in case we happen to be in
the tropics, where the sun may pass between the zenith and
the celestial pole. Even in temperate latitudes a celestial
body may do this, if we happen to observe a star instead of
the sun. In such a case, if the ship is in the northern
hemisphere, the navigator will observe the sun's altitude
toward the north at noon instead of toward the south, as
usual. Furthermore, in very high northern latitudes, the
"midnight sun," as it is called, can be observed toward the
north, and below the celestial pole. This is the minimum
altitude during the day, instead of the maximum ; but it is
usable for a latitude determination. Such an observation is
called a "lower transit" ; and it can often be observed in the
case of stars in temperate latitudes.
If we now remember to call northerly latitudes and
declinations plus, and southerly ones minus, we have the
following complete set of formulas for the present problem,
including observations in both hemispheres. These formulas
are so arranged that we can easily choose the right formula,
by having regard to the + and — signs. But the right
formula once chosen, the latitude is calculated without
marking declinations with either the + or — sign.
if lat. greater than dee., lat. = 90° + dec. — alt. (1)
if dec. greater than lat., lat. = dec. + alt. - 90° (2)
lat.1 and
dee. both +
or both —
if lower transit, lat. = 90° + alt. - dec. (3)
lat. and dec., 1 lat = ^ _ alt _ dec (4)
one +, one — j
We shall now give some more examples ; and to enable
the reader to follow star observations correctly we reprint
part of the upper halves of pages 94 and 95 (our pp. 91, 92)
of the Nautical Almanac. These contain the right ascensions
and declinations (p. 85) of a quantity of bright stars for
various dates in the year. These numbers are correct for the
moment of "upper transit," which is the moment when these
1 Latitude and declination are abbreviated lat. and dec.
90 NAVIGATION
stars attain their maximum altitudes. This event cannot
be called a noon-sight in the case of a star ; but it is observable
in a manner perfectly similar to a solar noon-sight.
These stellar right ascensions and declinations change
so slowly that it is unnecessary to use interpolation when
taking them from the almanac pages.
Proceeding now to our examples, suppose that on shore,
at Sandy Hook Light, approximate latitude and longitude
40° 28' N., 74° 0' W., on Monday, Dec. 17, 1917, at noon, the
double altitude of the sun's lower limb was observed with a
sextant and artificial horizon, and found to be 51° 48'. The
index correction required by the sextant was + 4' ; and the
G. M. T. by chronometer was 4* 56TO at the moment the
observation was made. Find the latitude. We have :
Observed double altitude 51° 48' (1)
Index correction + 4 (2)
Adding (1) and (2) gives corrected double altitude 51° 52' (3)
Halving (3) gives observed altitude 25 56 (4)
Correction from Table 61 (p. 247) + 14^ (5)
Adding (4) and (5) gives fully corrected altitude 26° 10' (6)
Now use formula (4) (p. 89) because latitude is +
and declination is - . Write 90 0 (7)
Subtracting (6) from (7) gives 90° - corrected altitude . . 63 50 (8)
Interpolate declination from almanac (p. 76). This
gives declination 23 22 (9)
Subtracting (9) from (8) gives for the latitude 40 28 (10)
With regard to the foregoing example it is worth remark-
ing that if there had been no available chronometer set to
Greenwich time, it would still have been possible to calculate
the observation. For the known approximate longitude,
even if only a dead-reckoning (p. 5) longitude, would be
quite accurate enough to make possible the interpolation of
the declination from the almanac. And in the present
example, the chronometer was only used in getting the
declination printed in line (9) above.
1 Dip correction from Table 7 not needed because the artificial
horizon was used.
OLDER NAVIGATION METHODS
91
APPARENT PLACES OF STARS, 1917
From Nautical Almanac, p. 94
FOR THE UPPER TRANSIT AT GREENWICH
RIGHT ASCENSION
Mr*
CONSTELLA-
^
^
1-4
TH
M
'
,_,
rt
<N
CO
IN t/»
TION NAME
q
>>
O
S.
.
j
•
3
•
S
1
•3
1-3
8
<
I
1
1
1
1
h in
s
s
8
S
8
s
s
s
s
8
1
<* Androm.
0 4
6.3
6.4
7.4
8.4
9.4
10.0
10.3
10.3
10.0
9.6
2
ft Cassiop.
0 4
44.8
44.4
45.7
47.3
48.7
49.7
50.1
49.9
49.3
48.4
3
0Ceti
039
26.5
26.3
27.0
28.0
28.9
29.7
30.0
30.1
29.8
29.5
4
& Cassiop.
1 20
23.9
22.3
23.5
25.1
26.7
28.1
28.9
29.2
29.0
28.2
5
«Urs. Min.
1 29
89.0
22.9
45.5
77.6
112.8
142.4
161.2
166.4
155.3
129.0
6
a Eridani
1 34
39.1
36.8
37.6
38.8
40.3
41.5
42.3
42.4
41.9
41.1
7
a. Arietis
2 2
31.0
30.1
30.8
31.7
32.7
33.6
34.3
34.6
34.7
34.5
8
9 Eridani
255
8.8
6.8
7.2
7.9
9.0
10.0
10.8
11.3
11.4
11.0
9
a. Persei
3 18
25.9
23.9
24.4
25.5
26.8
28.2
29.3
30.2
30.6
30.5
10
a Tauri
431
11.7
10.3
10.5
11.0
11.9
12.8
13.7
14.5
15.0
15.2
11
|3 Orionis
5 10
35.1
33.7
33.7
34.2
34.7
35.6
36.5
37.3
37.8
38.1
12
a- Aurigse
5 10
36.5
34.5
34.6
35.2
36.2
37.5
38.7
39.9
40.7
41.1
13
y Orionis
5 20
43.1
41.7
41.7
42.1
42.8
43.7
44.6
45.4
46.0
46.4
14
f Orionis
532
2.4
1.0
1.0
1.3
2.0
2.8
3.7
4.5
5.2
5.5
15
a Orionis
550
43.1
41.8
41.7
42.0
42.7
43.5
44.4
45.3
46.0
46.4
16
a Argus
622
9.2
6.1
5.5
5.4
6.0
6.9
8.1
9.3
10.2
10.6
17
a Can. Maj.
641
31.6
30.2
30.0
30.1
30.6
31.3
32.2
33.1
33.8
34.3
18
eCan. Maj.
655
24.1
22.6
22.2
22.2
22.6
23.3
24.2
25.2
26.0
26.5
19
a Can. Min.
734
59.7
59.0
58.7
58.8
59.1
59.8
60.5
61.5
62.3
63.0
20
ft Gemin.
740
17.1
16.3
16.0
16.0
16.4
17.1
18.0
19.0
20.0
20.8
21
< Argus
820
51.4
49.0
48.0
47.3
47.2
47.8
48.9
50.4
51.8
52.8
22
* Argus
9 4
58.6
57.9
57.3
56.9
56.8
57.1
57.8
58.9
60.1
61.0
23
ft Argus
9 12
20.6
18.1
16.4
15.1
14.5
14.8
16.0
17.9
20.0
21.7
24
a Hydrse
9 23
32.5
32.6
32.2
32.0
32.0
32.3
32.9
33.7
34.7
35.6
25
a Leonis
10 3
59.2
59.7
59.3
59.1
59.0
59.2
59.7
60.5
61.4
62.4
Had it been thus necessary to get the declination without
using the chronometer, we should have proceeded as follows :
Apparent solar time of noon (p. 75) 0* Om (1)
Approximate longitude = 74° 0' W. = (at 15° to
the hour) 4 56 W. (2)
Adding (1) and (2) (p. 81) gives approximate
Greenwich apparent time 4 56 (3)
Approx, eq. of time, Dec. 17, at 4* 56W (p. 76) + 4 (4)
Subtracting l (4) from (3) gives approximate
G. M. T 4 52 (5)
Declination interpolated for G. M. T. in line (5) is - 23° 22' (6)
1 The equation is additive to G. M. T., according to the note at
the foot of p. 76, and therefore to be subtracted from Greenwich
apparent time.
92
NAVIGATION
APPARENT PLACES OF STARS, 1917
From Nautical Almanac, p. 95
FOB THE UPPER TRANSIT AT GREENWICH
DECLINATION
No.
.
,H
^
rt
rt
M
IN
CO
SPECIAL NAME
MAO.1
a
a
<->
1
3
%
ft
<J
i
%
0
1
o
Q
*
1
0
+ 28
38.2
38.1
38.0
38.0
38.0
38.4
38.5
38.5
/
38.5
Alpheratz
2.2
2
+ 58
41.9
41.8
41.7
41.6
41.5
42.0
42.1
42.2
42.2
Caph
2.4
3
- 18
26.5
26.5
26.5
26.4
26.3
26.0
26.1
26.2
26.2
Deneb Kaitos
2.2
4
+ 59
48.7
48.7
48.6
48.4
48.3
48.6
48.8
48.9
49.0
Ruchbah
2.8
5
+ 88
52.2
52.2
52.1
52.0
51.8
52.0
52.2
52.4
52.5
Polaris
2.1
6
-57
39.7
39.7
39.6
39.4
39.2
39.0
39.2
39.3
39.4
Achernar
0.6
7
+ 23
4.5
4.4
4.4
4.3
4.3
4.6
4.7
4.7
4.7
Hamal
2.2
8
-40
38.3
38.3
38.3
38.2
38.1
37.7
37.8
38.0
38.1
Acamar
3.0
9
+ 49
34.3
34.3
34.3
34.2
34.1
34.3
34.3
34.4
34.5
1.9
10
+ 16
20.7
20.7
20.7
20.7
20.7
20.8
20.8
20.8
20.8
Aldebaran
1.1
11
- 8
17.8
17.8
17.9
17.9
17.8
17.5
17.6
17.7
17.7
Rigel
0.3
12
+ 45
55.0
55.1
55.1
55.1
55.0
54.9
54.9
55.0
55.1
Capella
0.2
13
+ 6
16.6
16.5
16.5
16.5
16.5
16.7
16.7
16.6
16.6
Bellatrix
1.7
14
- 1
15.2
15.3
15.3
15.3
15.3
15.0
15.1
15.1
15.2
Alnitam
1.8
15
+ 7
23.6
23.5
23.5
23.5
23.5
23.7
23.7
23.6
23.6
Betelgeux
1.0-1.4
16
-52
39.0
39.2
39.3
39.3
39.2
38.7
38.7
38.9
39.1
Canopus
-0.9
17
- 16
36.1
36.2
36.3
36.3
36.3
35.9
36.0
36.1
36.2
Sirius
- 1.6
18
-28
51.5
51.7
51.7
51.8
51.7
51.3
51.4
51.5
51.6
Adhara
1.6
19
+ 5
26.3
26.2
26.2
26.2
26.2
26.3
26.2
26.2
26.1
Procyon
0.5
20
+ 28
13.6
13.6
13.6
13.7
13.7
13.5
13.5
13.4
13.4
Pollux
1.2
21
-59
14.4
14.6
14.8
14.9
14.9
14.4
14.4
14.5
14.7
1.7
22
- 43
5.7
5.9
6.1
6.2
6.2
5.8
5.8
5.9
6.0
2.2
23
- 69
22.4
22.6
22.8
22.9
23.0
22.5
22.4
22.5
22.7
Miaplacidus
1.8
24
- 8
17.9
18.1
18.1
18.2
18.2
18.0
18.0
18.1
18.2
Alphard
2.2
25
+ 12
22.2
22.2
22.2
22.2
22.2
22.2
22.1
22.0
21.9
Regulus
1.3
1 When tlie number in this column is very small, and especially when it is minus,
the star is very bright.
It is further to be noted that as we can thus obtain the
approximate G. M. T., we really know in advance the approx-
imate moment when the observation should be made. So
it is unnecessary to get the sextant ready a long time before
the observation ; and it is, in fact, better to observe at the
proper predetermined approximate moment rather than to
wait for the maximum altitude (p. 86).
When the ship's position at noon can be predicted with fair
approximation, it is thus possible to have the declination and
other numbers for calculating the noon-sight also all ready
OLDER NAVIGATION METHODS 93
in advance, so that the latitude will be immediately available
when the noon altitude has been read from the sextant.
We shall now consider the following example : Off St.
Paul de Loando, West Africa, approximate latitude 8° 55'
south, approximate longitude 12° 55' east, both predicted
in advance by D. R. for noon on Monday, Dec. 31. The
altitude of the sun's lower limb is to be measured. Index
correction is — 5'. Height of eye, 26 ft.
To prepare for the observation, we have, as before :
Apparent solar time of noon 0* Om (1)
Approximate D. R. longitude = 12° 55' east = (at 15° to
the hour) 52 E. (2)
Subtracting (2) from (1) gives approximate Greenwich
apparent time, Dec. 30 23 8 (3)
Approximate equation of time, Dec. 30, at 23* 8W
(p. 76) - 3 (4)
Subtracting (4) from (3), having regard to — sign of
(4), gives approximate G. M. T 23 11 (5)
The navigator will then make the observation when the
G. M. T. is 23A 11TO, as indicated by the chronometer, duly
corrected for error and -rate. This would of course also be
noon, or the time when the sun attained its maximum altitude
for the day.
Now the dials of chronometers are always divided into
12 hours, like ordinary watches, although navigators count
time through 24 hours, as we have seen (p. 75). The
reason is that the dial would be overloaded with numbers
if there were 24 hour divisions. Therefore, when we speak
of the chronometer indicating 23* 11"*, it must be under-
stood that the actual chronometer indication, or "chro-
nometer face," as it is sometimes called, would really be
II71 llm ; only, the navigator would call it 23* llm, astronomic
time. In this manner civil time still forces its way into
navigation, by way of the chronometer face.
To make the observation at the prearranged G. M. T. by
chronometer it is not desirable to carry that instrument out
into the sunlight, where the observer stands. It is much
94 NAVIGATION
better for the navigator to use his watch, and to calculate in
advance the "watch time" of the observation. To do this,
it is merely necessary to compare the watch with the chro-
nometer, and thus ascertain how much the watch is slow or
fast of the chronometer. This amount is called "chro-
nometer minus watch" (abbreviated C. — W.) ; and when the
watch is fast of the chronometer, C. — W. is marked with the
minus sign.
To obtain the watch time for the observation, we subtract
C. — W. from the G. M. T. In the present case we will
suppose the watch was 47 m fast of the chronometer. Then
C. — W. = — 47m. To get the watch time for the observa-
tion we must subtract — 47m from 23A llm. Subtracting a
minus number is equivalent to addition ; and so the watch
time is 23* llm + 47m = 23* 58TO. The observation would
be made as nearly as possible 2m before noon, by the watch.
In this connection it also becomes of interest to inquire
how the navigator's watch happened to be 47m fast of the
chronometer. It is customary aboard ship to set the deck
and cabin clocks, and all watches, to the ship's local apparent
time once a day at least. To do this, we proceed as follows :
Take from chronometer the G. M. T., corrected for error and rate (1)
Apply to this G. M. T. the eq. of time, giving Green'h app. time (2)
Apply to (2) the approximate D. R. longitude, adding it if longi-
tude is E., which gives ship's apparent time (3)
And set the watch to the time (3).
An example of this proceeding can be had from the data on
p. 93. Suppose the watch was to be set; and the chro-
nometer time was 23* Oro. We should then prepare to set the
watch in about 5m, when the
G. M. T. by chronometer would be 23* 5"» (1)
Chronometer error (corrected for rate) say — 2 (2)
Corrected G. M. T. by chronometer, (1) +(2) 23 3 (3)
Equation of time (p. 93) — 3 (4)
Greenwich apparent time, (3) + (4) 23 0 (5)
Approximate longitude (p. 93) 52 E. (6)
Ship's apparent time, (5) + (6) 23 52 (7)
OLDER NAVIGATION METHODS 95
And the watch would be set to 23* 52m, when the chro-
nometer face was 23A 5m ; or, which is the same thing, the
watch would be set at 8TO to 12 when the chronometer in-
dicated 5 minutes past 11.
Sometimes the navigator wishes the watch to be correct
by ship's apparent time at noon, but desires to set it right
half an hour sooner, so as to be free at noon to make an
observation. In that case he calculates by D. R. what the
longitude will be at noon, and proceeds practically in the
same way as before.
Resuming now the example of p. 93, we are still
off St. Paul de Loando, and at 2W before noon by the
watch (p. 94) the altitude of the sun's lower limb was
measured.
Suppose it was found to be 75° 34' (1)
The index correction was — 5 (2)
Adding (1) and (2), with regard to sign of (2), gives
corrected altitude 75 29 (3)
Correction from Table 6 +16 (4)
Correction from Table 7, for 26 ft. height of eye — 5 (5)
Adding (3), (4), (5) gives corrected altitude 75 40 (6)
Formula (2), p. 89, is the proper one, and the inter-
polated declination, disregarding sign, is 23 8 (7)
Latitude, by formula, is (6) + (7) - 90°, or 8 48 (8)
The latitude of the ship is therefore 8° 48' south, from the
above noon-sight observation. The difference of 7' from
the approximate latitude (p. 93) might easily be caused by
ocean currents.
Our next example is a star observation. Position of ship
by D. R. March 23, 1917, at 6* 3(T ship's time is : latitude
40° 25' N., longitude 46° 52' W., so that she is near the turning
point in the southern "lane route" followed by steamships
bound from New York to Fastnet in summer. The upper
transit (p. 89) of Sirius was observed; and the sextant
altitude was 33° 7'. Index correction, — 7' ; height of eye,
24ft.
96 NAVIGATION
The calculation is as follows :
Observed altitude of Sirius 33° 7' (1)
Index correction — 7 (2)
Adding (1) and (2), having regard to minus sign of (2),
gives corrected altitude 33 0 (3)
Correction Tables 6 and 7, combined — 6 (4)
Adding (3) and (4) gives finally corrected altitude .... 32 54 (5)
Use formula (4), p. 89, because latitude is + and decli-
nation of Sirius -. We have 90° (6)
Subtract (5) from (6), giving (90° - altitude) 57 6 (7)
Declination of Sirius (p. 92), disregarding sign, is. . . 16 36 (8)
Subtract (8) from (7), giving (90°— altitude —declina-
tion), or the latitude 40 30 (9)
Ship's latitude at the moment of observation was therefore
40° 30' N.
In making such a star observation, it is of course possible
to follow the star with the sextant until it begins to
dip (p. 86) toward the horizon exactly as we have ex-
plained for the sun. But it is preferable to prepare for the
observation in advance, and to make it at a definite prede-
termined minute by the navigator's watch. To make such
preparation, it is necessary to use pages 96 and 97 of the
Nautical Almanac, parts of which pages are reprinted here
(pp. 97, 98).
The almanac page 96 gives for all the bright stars the
G. M. T. of upper transit (p. 89) at Greenwich, for the first
day of each month. And it will be noticed that the upper
transit is here called "meridian transit," which is practically
another name for the same thing. Almanac page 97 (our
p. 98) then gives a subtractive correction, applicable to the
numbers on page 96, to make them correct on days of the
month other than the 1st.
Another small correction is still required to make the
numbers right in the approximate D. R. longitude of the ship,
instead of the longitude of Greenwich, as used on almanac
page 96. This correction is subtractive, if the ship is in west
longitude, and additive, if she is in east longitude ; and the
OLDER NAVIGATION METHODS
97
MERIDIAN TRANSIT OF STARS, 1917
From Nautical Almanac, p. 96
GREENWICH MEAN TIME OF TRANSIT AT GREENWICH
CONSTELLA-
TION
NAME
MAO.
z
3
t
h
3
S3
g
•<
>•
<!
S
fc
H
CO
g
O
o
Z
I
Q
h m
h m
h m
h m
h m
h in
h m
h m
h m
a Androm.
2.2
5 21
3 19
1 29
23 23
21 25
13 22
11 24
9 22
7 24
ft Cassiop.
2.4
5 22
3 20
1 30
23 24
21 26
13 22
11 24
9 22
7 24
PCeti
2.2
5 56
3 54
2 4
f o a!
(23 SSS
22 0
13 57
11 59
9 57
7 59
S Cassiop.
2.8
6 37
4 35
2 45
0 43
22 41
14 38
12 40
10 38
8 40
a Urs. Min.
2.1
6 47
4 45
2 54
0 52
22 50
14 49
12 51
10 49
8 51
a Eridani
0.6
6 51
4 49
2 59
0 57
22 55
14 52
12 54
10 52
8 54
a Arietis
2.2
7 19
5 17
3 27
1 25
23 23
15 20
13 22
11 20
9 22
6 Eridani
3.0
8 12
6 10
4 20
2 18
0 20
16 12
14 14
12 12
10 14
a Persei
1.9
8 35
6 33
4 43
2 41
0 43
16 35
14 38
12 36
10 38
a Tauri
1.1
9 47
7 46
5 55
3 54
1 56
17 48
15 50
13 48
11 50
ft Orionis
0.3
10 27
8 25
6 35
4 33
2 35
18 27
16 29
14 28
12 30
a Aurigse
0.2
10 27
8 25
6 35
4 33
2 35
18 27
16 29
14 28
12 30
y Orionis
1.7
10 37
8 35
6 45
4 43
2 45
18 37
16 39
14 38
12 40
e Orionis
1.8
10 48
8 46
6 56
4 54
2 56
18 49
16 51
14 49
12 51
a Orionis
1.0-1.4
11 7
9 5
7 15
5 13
3 15
19 7
17 9
15 7
13 9
a Argus
-0.9
11 38
9 36
7 46
5 44
3 46
19 39
17 41
15 39
13 41
a Can. Maj.
- 1.6
11 57
9 55
8 5
6 3
4 5
19 58
18 0
15 58
14 0
e Can. Maj.
1.6
12 11
10 9
8 19
6 17
4 19
20 12
18 14
16 12
14 14
o Can. Min.
0.5
12 51
10 49
8 59
6 57
4 59
20 51
18 53
16 52
14 54
ft Gemin.
1.2
12 56
10 54
9 4
7 2
5 4
20 57
18 59
16 57
14 59
e Argus
1.7
13 36
11 34
9 44
7 42
5 44
21 37
19 39
17 37
15 39
A Argus
2.2
14 20
12 19
10 28
8 27
6 28
22 21
20 23
18 21
16 23
ft Argus
1.8
14 28
12 26
10 36
8 34
6 36
22 28
20 30
18 28
16 31
a Hydras
2.2
14 39
12 37
10 47
8 45
6 47
22 40
20 42
18 40
16 42
a Leonis
1.3
15 19
13 17
11 27
9 25
7 27
23 20
21 22
19 20
17 22
amount of it is 10* for every 15° in the ship's longitude.
After it has been applied, the result will be the ship's mean
solar time of the star's upper transit.
As an example, let us take the preparation for the fore-
going observation of Sirius, or a Can. Maj. We have :
G. M. T. of upper transit, March 1, from almanac
page 96 above 8* 5"* (1)
Correction for 23d day of month, from almanac
page 97 (our p. 98) - 1 27 (2)
Correcting (1) with (2), having regard to - sign of (2) 6 38 (3)
Further correction for longitude 46° 52' W., at 10* per
15° of longitude, approximately , 1 (4)
Subtracting (4) from (3) gives ship's mean solar time
of the observation 6 37 (5)
98
NAVIGATION
MERIDIAN TRANSIT OF STARS, 1917
From Nautical Almanac, p. 97
CORRECTIONS TO BE APPLIED TO THE MEAN TIME OF TRANSIT ON
THE FIRST DAY OF THE MONTH, TO FIND THE MEAN TIME OF
TRANSIT ON ANY OTHER DAY OF THE MONTH
DAY OF
MONTH
CORRECTION
DAY OF
MONTH
CORRECTION
DAY OF
MONTH
CORRECTION
h m
h m
h m
1
-0 0
11
-0 39
21
-1 19
2
0 4
12
0 43
22
1 23
3
0 8
13
0 47
23
1 27
4
0 12
14
0 51
24
1 30
5
0 16
15
0 55
25
1 34
6
-0 20
16
-0 59
26
-1 38
7
0 24
17
1 3
27
1 42
8
0 28
18
1 7
28
1 46
9
0 31
19
1 11
29
1 50
10
0 35
20
1 15
30
1 54
11
-0 39
21
- 1 19
31
- 1 58
NOTE. If the quantity taken from this Table is greater than the
mean time of transit on the first of the month, increase that time
by 23" 56m and then apply the correction taken from this Table.
The actual observation was made at 6A 30m, ship's time,
as indicated by the navigator's watch. The difference of
7m between 6* 30", and 6* 37m in line (5) above, is due to the
equation of time (p. 77), which is 7™ on March 23. This
7m, if applied (with its proper sign from the almanac) to
line (5) above, will give the ship's apparent time; and we
have seen that watches and clocks on board are usually
kept set to apparent and not mean ship's time (p. 94).
To complete this part of our subject, we have still to con-
sider a few additional points of interest. For instance, a
star chosen for observation may be one of the planets :
Mars, Jupiter, or Saturn. These look like very bright stars
in the sextant telescope; and calculations depending on
them are similar to those described for stars. The planetary
declinations and the G. M. T.'s of their upper transits are
given in the almanac, but not on the pages reprinted here.
OLDER NAVIGATION METHODS 99
The moon is now so rarely observed that we have not given
examples of lunar observations.
Sometimes an "ex-meridian" observation of the sun or
a star is made at a time very near the upper transit, on a
day when the actual transit observation could not be secured
because of clouds. There are special tables 1 for calculating
observations of this kind; but we have not included them
here because all such observations can be satisfactorily
treated by a new general method to be explained later
(p. 108).
Having now fully treated the older standard method of
determining the ship's latitude, let us next consider the older
way of obtaining the longitude. This cannot be done when
the sun (or a star) is near its maximum altitude, as already
explained (p. 88). The most favorable opportunity occurs
when the observed object bears (p. 44) east or west; but
it is not always possible to get the observation on such a
bearing. In that case, the longitude observation, often
called a "time-sight," must be taken when the sun is near
the desired bearing, but always avoiding, if possible, observa-
tions at very low altitudes. And if a very low altitude has
been observed in an emergency, it can sometimes be checked
by a later observation at a better altitude.
The principle on which the time-sight depends is simple.
Calculations based on the measured altitude make known
the ship's mean time at the moment of observation. At
the same moment the chronometer face (p. 93), duly cor-
rected for error and rate, tells us the G. M. T. The
difference between the two times then gives us the longitude
(see p. 82).
The calculations for this problem are made by means of
Table 4 (trigonometric logarithms) and Table 10 ("haver-
sines"). These haversines (abbreviated hav.) are really
additional trigonometric logarithms; and Table 10 gives
in every case not only the haversine itself, which is really
1 Tables 26 and 27 of Bowditch's "Navigator," for instance.
100 NAVIGATION
a logarithm, but also, in the adjoining heavy type col-
umns, the number (abbreviated No.) of which the haver-
sine is the log. This additional heavy type number is not
given throughout the entire table, but only when necessary
for working Sumner line calculations (see Chapter IX,
p. 108). It is not needed in working time-sights.
The argument (p. 10) of the haversine table is a double
argument, not to be confounded with the pairs of arguments
already explained (p. 11). In the haversine table, the argu-
ment is generally given in degrees and minutes, as well as
(for convenience) in hours and minutes of time, allowing
the usual 15° to each hour, etc.
We shall now solve our time-sight problem for the sun;
and in doing so shall make use of two angles not hitherto
employed: the "polar distance" (abbreviated p), and the
"half sum" (abbreviated s). We shall also, for brevity,
indicate the ship's apparent solar time by T. Then we
have the following formulas :
If lat. and dec. are both + or both — . . p = 90° — dec. (1)
If lat. and dec. are one + and one — . . . p = 90° + dee. (2)
In every case s = % (alt. + lat. + p) (3)
If time-sight was made before noon, ship's time,
hav. (24* — T) = sec lat. + esc p + cos s + sin (s — alt.) (4)
If time-sight was made after noon, ship's time,
hav. T= sec lat. + esc p + cos s + sin (s — alt.) (5)
In using these formulas, we have to choose between (1)
and (2), and also between (4) and (5). Formula (3) is
always used. No attention need be given to the signs
of the declination or latitude except in choosing between
formulas (1) and (2) for calculating p; and in choosing
between (4) and (5), we have merely to note whether the
time-sight was taken in the forenoon or afternoon by ship's
time.
We also desire to emphasize especially that these formulas
presuppose the latitude to be known. This is merely
another application of the principle (p. 88) that both lati-
OLDER NAVIGATION METHODS 101
tude and longitude cannot be determined from a single
observation. It follows that in using this method we must
first determine the latitude by a noon-sight before we can
calculate the time-sight for longitude. If the time-sight
was taken in the afternoon, the noon-sight will naturally
have preceded it, and the ship's latitude at noon will be
known. This noon latitude must then be carried forward
to the moment of the afternoon time-sight by D. R. methods
(p. 7) ; and the latitude thus obtained must be used for
calculating the time-sight.
But if the time-sight was a forenoon observation, it cannot
be properly calculated until noon, when the latitude will
be determined. After that, the latitude can be carried
backwards by D. R. to the moment of the forenoon time-
sight, and the latter can be calculated.
But if the navigator, because of emergency, needs his
longitude at once, after taking the forenoon time-sight, he
must obtain the latitude by a D. R. calculation based on the
last good noon-sight. Most navigators calculate morning
time-sights in this way, and then repeat the calculation
after the new noon-sight has been obtained. The latter
calculation will be preferable to the former, because the
further the latitude is carried along by D. R., the less accurate
will it be. And any error in the latitude used in the calcula-
tion will impress a consequent error on the calculated longi-
tude.
We shall now work some time-sight examples. On board
ship, at sea, Dec. 18, 1917, in the afternoon, D. R. latitude
42° 20' N., D. R. longitude 35° 16' W., the altitude of sun's
lower limb was observed to be 14° 19'. The time was taken
with the navigator's watch, and was 2h 29m 58*. A com-
parison of the watch and ship's chronometer gave C. — W. =
2h 27m 8*. The chronometer correction was 2m 8* slow of
G. M. T. The index correction of the sextant was + 4' ;
height of eye, 24 ft. Calculate the ship's longitude.
We have first to find, for the moment of the observation.
102 NAVIGATION
values of the declination and equation of time. To do this,
we have :
Watch time of observation 2» 29"» 58« (1)
C. -W 2 27 8 (2)
Adding (1) and (2) gives chronometer time of
observation 4 57 6 (3)
Chronometer correction, slow 2 8 (4)
Adding (3) and (4) gives G. M. T. of observation 4 59 14 (5)
For the G. M. T. (5) we interpolate the declina-
tion (p. 76), finding - 23° 24' (6)
and for the same G. M. T. we interpolate the
equation of time + 3W 21* (7)
Now, adding (5) and (7) gives Greenwich ap-
parent time of observation 5* 2m 35» (8)
Next we inspect the formulas (p. 100), choosing (2) be-
cause latitude is + and declination — , and (5) because the
sight was an afternoon one.
We now have, from line (6), declination (disregard-
ing sign) 23° 24' (9)
to which, by formula (2), we add , 90 0 (10)
giving p 113 24 (11)
The observed altitude was 14 19 (12)
Index correction +4 (13)
Adding (12) and (13) gives corrected altitude 14 23 (14)
Correction, Table 6 +12 (15)
Correction, Table 7 - 5 (16)
Adding (14), (15), (16) gives finally corrected altitude 14 30 (17)
The latitude by D. R. is 42 20 (18)
Adding (11), (17), (18) gives ' 170 14 (19)
Halving (19) gives (by formula (3), p. 100) s 85 7 (20)
Subtracting (17) from (20) gives (s - alt.) 70 37 (21)
Next we apply formula (5), p. 100. We have:
sec lat. (18) from Table 4, page 238 0.13121 (22)
esc p (11) from Table 4, page 219 0.03727 (23)
cos s (20) from Table 4, page 200 8.93007 (24)
sin (s - alt.) (21) from Table 4, page 215 9.97466 (25)
sum (22) to (25) = hav. T, by formula (5) 9.07321 ' (26)
1 This sum has been diminished by 10 arbitrarily (see p. 25),
which must always be done when the sum of logs is larger than 10.
OLDER NAVIGATION. METHODS 103
TV corresponding to (26) from Table 10, page 260, is 2* 40W 59* (27)
Greenwich apparent time (8) by watch and
chronometer is 5 2 35 (28)
Subtract (27) from (28), giving time difference
between ship and Greenwich 2 21 36 (29)
Turning (29) into degrees with Table 9, page 249,
gives 35° 24' W. (30)
and (30) is the ship's longitude from this time-sight.
Upon comparing the D. R. longitude (35° 16' W.) with the
result of the time-sight (35° 24' W.), we find that the ship
is 8' west of her D. R. position. This means, of course, that
there has been a westerly "set" of current in the interval
between the last accurate determination of longitude and
the present one. It would be proper for the navigator to
calculate from this the amount of westerly drift per hour,
and to allow for it in carrying forward his longitude by D. R.
from the present time-sight. It is also clear that the
northerly or southerly set of the current can be similarly
measured and allowed for by comparing the D. R. latitude
with the latitude from a noon-sight (cf. p. 95). It is the
general custom of navigators to ascribe such differences to
ocean currents, never to uncertainty in the astronomic results.
Dead reckoning is never allowed any weight as against a
sextant observation.
The reader will have noticed that the foregoing calculation
has been made in great detail, so that a beginner may have
no difficulty in understanding it. But a practiced navigator
would of course work the calculation in a much more con-
densed form, in such a way as to bring the logarithms next
to the numbers to which they belong. We shall therefore
now repeat the same example in such a condensed form :
1 If the observation had been made before noon, we should have
used formula (4) and should here have obtained 24* — T, instead
of T. This 24* - T would then be subtracted from 24*, to get
T, before continuing the calculation. Thus the form of calculation
would contain another line between (27) and (28), in the case of
a forenoon observation.
104
NAVIGATION
TIME-SIGHT, CONDENSED FORM. SUN
Watch time :
C. - W. :
Chr. time :
Chr. corr'n :
2» 29»» 58' (1)
2 27
4 57
+ 2
6
8
14
21
35
G. M. T. : 18«> 4 59
Eq. of time : + 3
G. app. time : 5 2
Decl. 18th, 4* : 23° 23'.7
H. D.: 0.1
Decl. 4» 59™ : 23 24
p: 113 24
(2)
(3)
(4)
(5)
(7)
(8)
Obs'd alt. :
14° 19' (12)
Index :
+ 4 (13)
Table 6 :
+ 12 (15)
Table 7 :
- 5 (16)
Corr'd alt. :
14 30 (17)
(6)
(11)
Eq. time, 18th, 4* : + 3m 22*.3
H. D.: 1.2
Eq. time, 4s 59™ : +3 21.1
(7)
Corr'd alt.
Lat., D. R.
p:
sum of 3 :
s :
s — alt. :
By chron.,
r 14° 30' (17)
: 42 20 (18) sec lat. :
113 24 (11) esc p:
0.13121 (22)
0.03727 (23)
8.93007 (24)
: 9.97466 (25)
2)170 14 (19)
85 7 (20) cos s :
70 37 (21) sin (s-alt.):
sum of 4 :
T = ship's app. time :
Greenwich app. time :
Longitude :
or:
9.07321 (26) = hav. T
(or 24* - T) *
2» 40" 59* (27)
5 2 35 (8)
2ft 21"» 36* (29)
35° 24' W. (30)
When the object observed is a star or planet, the choice
between formulas (4) and (5), p. 100, is not quite the same
as in the case of a solar time-sight. We must use (4) if there
is any east in the star's bearing at the moment of observation ;
and (5), if there is west in the bearing. The more nearly the
star bears due east or west, the more accurate will be the
resulting longitude. The use of formulas (1), (2), and (3)
is the same as for the sun ; but T, in the case of a star, is no
longer the ship's apparent solar time. Instead, it is called
1 See p. 103, footnote.
OLDER NAVIGATION METHODS 105
the star's "hour-angle." To get the longitude, we must
first (p. 85) calculate the Greenwich sidereal time corre-
sponding to the G. M. T. of the observation, as taken from
the chronometer, duly corrected for error and rate; and
then use the following formulas :
(6) Greenwich sid. time1— right-ascension of star = Greenwich
hour-angle.
,„ | West long. = Greenwich hour-angle - T,
\ East long. = T — Greenwich hour-angle.
As an example of a star observation we shall take the
following :
At sea, just before sunrise, Dec. 17, 1917, off Cape Agulhas,
latitude by D. R. 35° 20' S., longitude by D. R. 20° 41' E.,
the altitude of Sirius was measured, and found to be 40° 3'.
The star bore west, and the height of eye was 22 ft. Index
correction was -f 5'. Time by watch, 16* 29"1 48*, or 4* 29"*
48' A.M., civil time, Dec. 18; C. - W., - lh 2Zm 50'; chro-
nometer fast of G. M. T. 2m 28*.
The calculation would proceed thus:
Watch time of observation 16* 29"» 48* (1)
C. - W - 1 23 50 (2)
Adding (1) and (2), having regard to — sign of (2),
gives chronometer time of observation 15 5 58 (3)
Chronometer correction, fast — 2 28 (4)
Adding (3) and (4), having regard to - sign of (4),
gives G. M. T. of observation 15 3 30 (5)
Right ascension mean sun, Greenwich mean noon,
Dec. 17 (p. 83) 17 42 10 (6)
Correction for " time past noon " (see p. 84) .... 2 28 (7)
Adding (6) and (7) gives right ascension of mean
sun 17 44 38 (8)
Adding (5) and (8) (see p. 85) gives Greenwich
sidereal time of the observation 8 l 48 8 (9)
Right ascension of Sirius, Dec. 17, is (p. 91) .... 6 41 34 (10)
Subtracting (10) from (9) gives Greenwich hour-
angle (formula (6), above) 2 6 34 (11)
1 24A may always be added or dropped here, if necessary.
106 NAVIGATION
Next we calculate T by formula (5), p. 100. We have:
Declination of Sinus, Dec. 17 (p. 92) - 16° 36' (12)
By formula (1), p. 100, subtract (12) from 90°,
without attention to sign of (12), giving p. . 73 24 (13)
The observed altitude was 40 3 (14)
The index correction was +5 (15)
Table 6 correction - 1 (16)
Table 7 correction — 5 (17)
Adding (14), (15), (16), (17), having regard to
signs, gives corrected altitude 40 2 (18)
The latitude by D. R. was 35 20 (19)
Adding (13), (18), and (19) gives 148 46 (20)
Halving (20) gives s. 74 23 (21)
Subtracting (18) from (21) gives (s - altitude) . . 34 21 (22)
Now applying formula (5), page 100, we have :
sec latitude (19) from Table 4, page 231 0.08842 (23)
esc p (13) from Table 4, page 212 0.01849 (24)
cos s (21) from Table 4, page 211 9.43008 (25)
sin (s - altitude) (22) from Table 4, page 230 9.75147 (26)
Summing (23) to (26) gives hav. T, by form. (5) . . 9.28846 1 (27)
712 corresponding to (27), from Tab. 10, p. 263 is . . 3* 29m 14* (28)
Difference between (28) and (11) is the longi-
tude by formula (7), page 105 1 22 40 E. (29)
Turning (29) into degrees with Table 9, page
249, gives 20° 40' E. (30)
The D. R. longitude, 20° 41' E., was therefore within 1' of
the longitude from this time-sight, and this shows that the
ship has not been affected by ocean currents since the last
observation. It is also interesting to note how near sunrise
the observation was made. The twilight must have been
quite strong, and the star therefore dim. But star observa-
tions can be made best in twilight because the horizon line
can then be seen distinctly.
1 This sum has also been diminished by 10 (see footnote, p. 102).
2 Might be 24* — T, if the star bore E. instead of W. (see footnote,
p. 103).
OLDER NAVIGATION METHODS
107
The foregoing example can of course also be arranged in
condensed form, as follows :
TIME-SIGHT, CONDENSED FORM. STAR
Watch time :
C. - W. :
Chr. time :
Chr. eorr'n :
G. M. T. :
R. A. mean sun :
Corr'n, past noon :
Greenw'h sid. time :
R. A. of Sirius :
Greenwich hour-ang.
T., from (27) :
Long. :
or:
R. A. of Sirius :
Dec. of Sirius :
p:
sec lat. :
esc. p :
cos s:
sin (s — alt.) :
sum of 4 :
16* 29" 48' (1)
Obs'dalt. : 40° 3'
(14)
-1
23
50 (2)
Index :
+ 5
(15)
15
5
58 (3)
Table 6 :
-1
(16)
- 2
28 (4)
Table 7 :
-5
(17)
15
3
30 (5)
Corr'd alt. : 40
2
(18)
17
42
10 (6)
Lat. D. R. : 35
20
(19)
2
28 (7)
p: 73
24
(13)
8
48
8 (9)
sum: 2)148
46
(20)
6
41
34 (10)
s: 74
23
(21)
: 2
6
34 (11)
(s - alt.) : 34
21
(22)
3
29
14 (28)
1
22
40 E. (29)
20°
40' E. (30)
6* 41™ 34'
(10)
- 16° 36'
(12)
73 24
(13)
0.08842
(23)
0.01849
(24)
9.43008
(25)
9.75147
(26)
9.28846 (27) = hav. T (or 24* - T)
Having now fully explained both the noon-sight and the
time-sight, we shall close this chapter with a strong recom-
mendation to young navigators to familiarize themselves with
the observation of stars. These always furnish a valuable
check on sun observations : and at times of danger may save
the ship when clouds have obscured the sun for days, and
clearing occurs after sunset. It is easy to learn to know the
principal stars from Jacoby's "Astronomy," Chapter III,
"How to Know the Stars."
1 See footnote, p. 103.
CHAPTER IX
NEWER NAVIGATION METHODS
THE reader may have noticed in Chapter VIII that there
is a very definite difference between the determination of
latitude by a noon-sight and longitude by a time-sight : for
the latitude is obtained without previous knowledge of the
longitude; but to get the longitude, a previous knowledge
of the latitude is essential. This is, of course, a decided
disadvantage in determining longitude, nor is there any
practicable direct way to get the longitude without first
knowing the latitude.
We have also seen (p. 101) that any existing uncertainty
in our knowledge of the latitude will produce an error in the
longitude computed from a time-sight. In situations of
danger it is important to ascertain how great this longitude
error may be. Suppose, for instance, we have calculated
a tune-sight with a D. R. latitude that we suspect may be
as much as 10' too small ; and we wish to know how much
our computed longitude may have been thereby put wrong.
The obvious way to find out is to recompute the longitude
with an assumed latitude 10' larger than the D. R. latitude.
The resulting longitude will then show the extreme range
of error that must have been produced if the D. R. latitude
was 10' too small.
A third calculation, with an assumed latitude 10' smaller
than the D. R. latitude, will similarly exhibit the extreme
possible range of longitude error in the other direction.
Thus these two extra calculations will show the limits of
longitude error that might be caused by a range of 20' in
the possible error of the D. R. latitude.
108
NEWER NAVIGATION METHODS 109
This rather obvious procedure was probably used long
ago by more than one intelligent navigator ; but it was first
published in 1837 by Thomas H. Sumner, an American
merchant captain. He used the method in dramatic cir-
cumstances of great danger ; and he brought his ship safely
into port. According to his own account, he made three
calculations of the longitude, using three assumed latitudes
differing by 10', and he of course obtained three different
longitudes. He then marked or plotted (p. 55) on his chart
the point indicated by the first assumed- latitude and its
computed longitude. At this point the ship must have been
located, if the first assumed latitude had been correct. The
other two latitudes, with their computed longitudes, indicated
two more points on the chart ; and at one of these points the
ship must have been, if either of these additional latitudes
was correct.
Sumner found that the three points on the chart lay in a
straight line; and it became at once evident that whatever
latitude he might assume (within reason) he would always
get a point on the same straight line, after computing the
longitude. In other words, although he did not know his
latitude accurately, and so could not compute his longitude
accurately, yet he had found a straight line on the chart
upon which his ship was surely situated.
Such a line can always be found in the way Sumner found
it, or in some preferable modern way; and such a line we
shall call a "Sumner line," though some writers on naviga-
tion prefer to call it a "line of position."
On the occasion of laying down his line, Sumner found that
it passed directly through Small's Light, near the Irish coast ;
and as the line bore E.N.E. on his chart, he simply put
the ship on that course, and in less than an hour he "made"
Small's Light, actually bearing E.N.E. £ E., and, as he says,
"close aboard." He had had no observations after passing
longitude 21° W., until the morning of Dec. 17, when these
historic events occurred. He was off a rocky lee shore, in
110 NAVIGATION
the midst of a winter gale, after crossing the Atlantic ; only
a seaman can understand the relief he must have felt when
that light suddenly appeared off the bow.
We have given this account of Sumner's experience to
impress on the young navigator that he must positively
familiarize himself with the Sumner method of navigation.
Should we be so fortunate as to have any experienced navi-
gator among our readers, we ask him to try the Sumner
method once more, in the manner explained below, even if
he may have found it troublesome in the past on account of
certain difficulties in its application. For the Sumner
method is the best method of navigation on all oceans and
at all times : even when a noon-sight is available for latitude,
it is better to treat it as a Sumner observation, and work
out the Sumner line.
The principal objection urged against it by certain prac-
tical navigators arises from the small scale of existing ocean
track charts, on which a distance of 10' is represented by
about -£ inch. A line like Sumner's, 20' long, would have
only a length of \ inch on the chart ; and such a little line
would not be long enough to show accurately the direction
in which it pointed. When near a coast, as in Sumner's
case, this difficulty disappears, because navigators always
have (or always should have and use) the large scale charts
that can be obtained for coastwise waters.
But it is inconvenient for navigators to begin using a
method off the coast, on the last day of a voyage, different
from the form employed for many days at sea. Therefore,
some authorities recommend the construction of a special
large scale chart, with its latitude and longitude lines, each
tune an observation is made throughout the voyage, so that
the Sumner line can always be drawn on a sufficiently large
scale. It is no wonder that navigators have not generally
adopted this somewhat laborious proceeding; and in the
method given below we shall utilize the Sumner idea without
requiring any lines to be drawn on charts.
NEWER NAVIGATION METHODS 111
Another objection to Sumner navigation is that it requires
too much calculation ; three longitude calculations for one
observation, as Sumner practiced it. This objection is also
quite removed now by the use of suitable tables such as we
give in the present volume.
But before proceeding to explain these tables, we must
outline briefly the real principle on which rests the com-
plete utilization of the Sumner method on the open sea.
There the navigator wants to know the ship's position in
both latitude and longitude; and will not be satisfied with
a mere line, with the ship "somewhere on the line." Along
the coast such a line might help him to find Small's Light ;
but he is not looking for coast lights at sea.
And the Sumner method takes care of this matter in the
simplest possible way. We have seen (p. 88) that two
different observations are always necessary by any method
to get both latitude and longitude. But two such observa-
tions by the Sumner method give two different lines on the
chart : arid as the ship must be located on both lines, her
actual position must be at their point of intersection. We
shall show how the required latitude and longitude of the
ship at the point of intersection can be found by a simple
calculation, without the drawing of any lines on the chart.
Coming now to the modern method of calculating a Sum-
ner line, we must first state a general fundamental principle
that may be easily verified by geometrical considerations.
The true bearing (p. 44) of a Sumner line on a chart is
always 90° greater than the true bearing or azimuth (p. 44)
of the sun (or star) at the moment of observation. Or, in
other words, the Sumner line bears at right angles to the
sun at the time of observation.
We shall show how the bearing or azimuth of the sun can
always be found from suitable "agimuth tables"; but the
Sumner line is not completely known from its bearing alone.
To locate it properly it is necessary to know in addition the
latitude and longitude of some point on the line, which we
112 NAVIGATION
will call a "Sumner point." Then, knowing such a point of
the line, and the bearing of the line, we may say we know the
line completely, and, if necessary, could draw it on a chart.
Now to find the required Sumner point. We always have
the D. R. position of the ship at the moment of observation ;
which we will call the "D. R. point." It is easy to find
out if the D. R. point is also a Sumner point. It is merely
necessary to calculate what the sun's altitude would be for
a ship at the D. R. point, and then compare this calculated
altitude with the one actually observed. If the D. R. point
was really a Sumner point (which will rarely happen), the
two altitudes will agree ; if not, the amount of disagreement
will show how far the D. R. point is distant from the nearest
Sumner point.1
The first step, then, in Sumner navigation, is the calcula-
tion of the altitude, supposing the ship to be at the D. R.
point at the moment of observation. To do this for a sun
observation, we first calculate the Greenwich apparent time
(abbreviated G. A. T.) of the observation, just as was done
in the case of a time-sight on p. 102. To this G. A. T. we
then add the ship's D. R. longitude, if east, or subtract it, if
west, to get T (p. 100), the ship's apparent time of the ob-
servation. We then use the formulas on p. 113, in which
X and Z are "auxiliary angles" required in the calculations,
but not otherwise of special interest. These formulas are
called the " cosine-haversine " formulas.
There are several other sets of formulas with which the
same problem can be solved. One set, called the " haversine "
formulas, involves the use of haversines only; another,
called the "sine-cosine" formulas, solves the problem with
sines and cosines. But neither is preferable to the following
cosine-haversine set.
1 This method is often called the Marcq Saint, Hilaire method ;
but it should probably be credited to Lord Kelvin, who published
" Tables for Facilitating Sumner's Method at Sea " in 1876. These
tables follow the method described above.
NEWER NAVIGATION METHODS 113
If observation was made before noon, ship's time,
hav. -X" = cos lat. + cos dec. + hav. (24* - T), (1)
If observation was made after noon, ship's time,
hav. X = cos lat. + cos dec. + hav. T, (2)
lat. — dec. = diff.1 of lat. and dec., if both are + or both — , (3)
lat. — dec. = sum1 of lat. and dec. if one is + and one — , (4)
No. hav. Z = No. hav. (lat. - dec.) + No. hav. X, (5)
Alt. = 90° - Z. (G)
Now we can compare the altitude computed by formula
(6) with the observed altitude, fully corrected for index
error, etc. The difference between the two altitudes in
minutes will be the distance in miles of the nearest Sumner
point from the D. R. point, for the minute and nautical
mile here correspond, as they do in the case of differences of
latitude (p. 15). The bearing of the Sumner point from the
D. R. point will be the same as the sun's azimuth if the ob-
served altitude is greater than the computed altitude : but if
the observed altitude is less than the computed, the bearing of
the Sumner point will be 180° greater than the sun's azimuth.
The bearing and distance of the Sumner point from the
D. R. point once known, it is easy, by means of the traverse
table (p. 10), to obtain the latitude and longitude of the
Sumner point from the known latitude and longitude of
the D. R. point ; or, which is the same thing, from the ship's
D. R. latitude and longitude.
Before giving examples of these calculations, it remains
to show how the sun's bearing or azimuth can be taken from
Table 11 (p. 284), called the azimuth table. The pair of
arguments (p. 11) for entering this table are: first, in the
left-hand column, the declination, which is here used without
regard to its sign; and second, in the four topmost hori-
1 In using formulas (3) and (4), pay no attention to + or —
signs after the right formula is once chosen. The difference between
latitude and declination is always taken by subtracting the smaller
from the larger ; and the sum by adding them, without regarding
their + or — signs. Cf. also p. 89.
114 NAVIGATION
zontal lines, T (p. 100), the ship's apparent time at the
moment of observation.
Having found this pair of arguments, we look in the
column under T, and in the horizontal line opposite the
declination. There we find an "index number." Next we
look up the altitude, as computed by formula (6), page 113,
in the right-hand column of the azimuth table, and follow
along the horizontal line belonging to that altitude, until
we reach a number equal (or nearly equal) to the index
number. Then we go down the column containing this
second appearance of the index number, and find the azi-
muth at the bottom of the page. The table gives approxi-
mate azimuths only, but the approximation is sufficient for
our present purpose.
The azimuths at the bottom of the page appear in four
horizontal lines, of which the upper two belong to forenoon
observations, and the lower two to afternoon observations.
All azimuths are counted from the north, through east,
south, and west, from 0° to 360°, like compass courses in
United States Navy practice (p. 41). It is important for
the navigator to record, at the time of observation, the word
"forenoon" or "afternoon," and also the sun's roughly
approximate bearing, to aid in choosing which of the azi-
muths at the bottom of the tabular page is the right one.
The record showing whether the observation was made in
the forenoon or afternoon limits the choice to two of the lines
of azimuths; and if there is any doubt remaining between
these two, the following rules may clear it up.
When latitude is + and declination — , azimuth is between
90° and 270°;
When latitude is + and declination +, if declination is
greater than latitude, azimuth is not between 90° and 270° ;
When latitude is — and declination — , if declination is
greater than latitude, azimuth is between 90° and 270° ;
When latitude is — and declination +, azimuth is not
between 90° and 270°.
NEWER NAVIGATION METHODS 115
In other cases, and especially when latitude and declina-
tion are nearly equal, the foregoing rules are insufficient, and
we must consult Table 12 (p. 290), the "auxiliary azimuth
table." This table has latitude and declination for its pair
of arguments, the former in the left-hand vertical column,
the latter in the topmost horizontal line : and in using the
table it is not necessary to pay attention to the + and —
signs of latitude and declination. Start with the latitude,
and follow its horizontal line to the right until you reach the
column having the declination at its head. There you will
find an "auxiliary angle," which must be compared with
the altitude computed by formula (6), page 113. Then :
If the computed altitude is greater than the auxiliary
angle, and if latitude is +, azimuth is between 90° and 270° ;
If the computed altitude is less than the auxiliary angle,
and if latitude is — , azimuth is between 90° and 270° ;
If the computed altitude is less than the auxiliary angle,
and if latitude is +, azimuth is not between 90° and 270° ;
If the computed altitude is greater than the auxiliary
angle, and if latitude is — , azimuth is not between 90° and
270°.
It will rarely happen that any of the foregoing rules will
be needed, if the navigator will make a careful observation
of the sun's azimuth with the azimuth circle or pelorus
(p. 44), as soon as possible after the sextant altitude has
been observed. The ship's course should also be specially
recorded when this observation is made. This proceeding
is not merely a convenience to avoid consulting the fore-
going rules in using the azimuth table : it is really essential
to safe navigation, for a comparison of the observed azi-
muth with that derived from the table will make the com-
pass error (p. 43) known. The variation is known from the
chart ; so that if we observe the compass error, we can allow
for the variation, and get the deviation. This can then be
compared with the deviation table (p. 48), to see if there has
been any change in the compass since leaving port. It is
116 NAVIGATION
a great advantage of the Sumner method that the sun's
azimuth comes out as a sort of by-product, so that the com-
pass can be verified without any additional special calcu-
lations.
We shall now illustrate all the above considerations by
means of examples ; beginning with the observation already
treated as a time-sight (p. 101). That observation we shall
now work by the Sumner method. From page 101 we take
the following :
Date of observation, Dec. 18, 1917, in the afternoon; D. R.
latitude, 42° 20' N. ; D. R. longitude, 35° 16' W. ; altitude observed,
14° 19' ; time by watch, 2* 29m 58' ; C. - W., 2* 27" 8« ; chronometer
correction, 2m S' slow of G. M. T. ; index correction, + 4' ; height of
eye, 24 ft.
From the preparatory part of the calculation (p. 102),
we also copy the following additional numbers :
Declination, line (6), page 102 -23° 24' (1)
Greenwich apparent time (G. A. T.) of observation,
line (8), page 102 5* 2- 35* (2)
We have next to calculate, by the formulas on page 113, the
altitude corresponding to the D. R. point, for which the
latitude and longitude are given above. The longitude is
35° 16' W., or, at 15° to the hour (Table 9, p. 249) :
D. R. longitude is 2* 21" 4* W. (3)
Subtracting (3) from (2), according to page 112,
gives ship's apparent time of observation, T. . 2 41 31 (4)
We are now prepared to apply formulas (1) to (6),
page 113. We choose formula (2) for an afternoon obser-
vation l ; and write :
1 For a forenoon observation we should choose formula (1), and
should therefore need to know 24* — T instead of T. This would
make necessary another line in the form of calculation, and it would
follow line (4). This new line might be numbered (4') ; and in it
would be written 24* — T, obtained by subtracting T (line 4) from
24*.
NEWER NAVIGATION METHODS 117
Cos lat., 42° 20' N. by D. R. (see Table 4, p. 238) .... 9.86879 (5)
Cos dec., 23° 24', line (1) (see Table 4, p. 219) 9.96273 (6)
Hav. T, 2* 41m 31', line (4) (see Table 10, p. 260) .... 9.07596 (7)
Adding (5) to (7) gives hav. X (dropping 20, p. 25) . . 8.90748 (8)
Now we choose formula (4), because latitude and declina-
tion are -+- and — ;
The latitude is, by D. R 42° 20' (9)
Adding (1) and (9) according to formula (4) gives
(lat. - dec.) 65° 44' (10)
Now we have, Table 10, page 266, No. hav. of (10) . . 0.29451 (11)
No. hav. X,1 line (8) 0.08082 (12)
Adding (11) and (12), according to formula (5), page
113, gives No. hav. Z 0.37533 (13)
And Z, corresponding to (13) is found from Table 10,
page 268 75° 34' (14)
Then, by formula (6) computed altitude =90° - Z (14),
or 14° 26' (15)
This computed altitude (15) must now be compared with
the observed altitude, fully corrected. We find :
Obs'd alt., fully corrected, line (17), page 102, is 14° 30' (16)
Difference between (15) and (16), in minutes, is the
distance of Sumner point from D. R. point in
miles (p. 113). It is 4 miles (17)
Next we must find the sun's azimuth from Table 11, page
286. The top argument for entering the table is T, line
(4), and it must be found in the "afternoon" lines. The
argument for the left-hand column is the declination, line (1).
Under T, and opposite declination, we find the tabular index
number 5872. 2 Then we find the computed altitude, line
(15), in the right-hand column of Table 11, page 286, and
1 This No. hav. X comes from Table 10, page 258, without looking
up the angle X at all. We simply find hav. X in the table, and take
the No. hav. X out of the adjoining heavy type column. No inter-
polations are needed, the nearest tabular numbers being sufficiently
accurate.
2 The index numbers and the azimuth need not be very accurate :
it is sufficient to use the nearest tabular arguments, so that inter-
polation is not essential.
118 NAVIGATION
follow its horizontal line till we again come upon the index
number 5872. It lies about halfway between 5703 and
5973. Going down the two columns containing these index
numbers, we find in the afternoon azimuth lines two values
of the azimuth, 217° and 323°. The choice between these
two numbers would be very easy, if the observer's record
contained even a rough estimate of the sun's bearing at the
time of observation. We have purposely not made this avail-
able, so as to show how to consult the directions on page
114, and there we find that when the latitude is -f and the
declination — , the azimuth is between 90° and 270°. So
we finally choose 217° for the sun's azimuth.
Since the observed altitude (16) is greater than the com-
puted altitude (15), the bearing of the Sumner point from
the D. R. point, according to page 113, is the same as the sun's
azimuth, or 217°. And as we now know the bearing and
distance of the Sumner point from the D. R. point, we can
find its latitude and longitude by a simple application of the
traverse table (p. 154).
We have merely to consider the bearing and distance to
be a course angle and distance, and imagine a ship to have
sailed from the one point to the other. In the present case,
the distance is 4 miles (line 17), the course 217° : and Table 1
(p. 164) gives the corresponding latitude 3'.2, departure 2.4.
The longitude difference is obtained from the departure by
Table 2 (p. 174) and is, for latitude 42°, about 3'.2. Drop-
ping odd fractions, the latitude difference and longitude differ-
ence both come out 3'. The Sumner point is therefore 3' dis-
tant from the D. R. point in both latitude and longitude.
And since the bearing 217° indicates on the compass card
that the Sumner point is south and west of the D. R. point,
it follows that :
Lat. of Sumner point = D. R. lat. — 3' =
42° 20' N. (line 9) - 3' 42° 17' N. (18)
Long, of Sumner point = D. R. long. +3' 35 19 W. (19)
Azimuth of Sumner line (p. Ill) 307° (20)
NEWER NAVIGATION METHODS
119
It is important for the reader to understand that the fore-
going calculation is given in extended detail so as to make
it easy for the beginner to follow. In condensed form,
we should have the following arrangement of the calculation,
corresponding to the condensed time-sight form (p. 104).
Part of the work here repeated from page 104 has no attached
reference numbers in parentheses : the new part of the work
has references to the detailed calculation just given.
SUMNER LINE, CONDENSED FORM. SUN
It.: 14° 19' Decl.4*: 23° 23'. 7 S.
+ 4 H. D. : 0.1
: + 12 Decl. 4* 59™ : 23° 24' S.
: - 5 Eq. time, 4*: +3«22«.3
It. : 14° 30' H. D. : 1.2
Eq. time, 4* 59»» : +3 21.1
Watch time :
2» 29" 58*
C. - W. :
2 27 8
Chr. time :
4 57 6
Chr. corr'n :
+ 28
G. M. T. 18th :
4 59 14
Eq. of time :
+ 3 21
G. app. time :
5 2 35
D. R. long. :
2 21 4W. (3)
Ship's app. time,
T: 2 41 31
(4)
hav. T (or 24* -T)
!; 9.07596
D. R. lat. :
42° 20' N.
(9)
cos lat. :
9.86879
Dec.:
23 24 S.
(D
cos dec. :
9.96273
sum = hav. X :
8.90748
No. hav. X :
0.08082 (12)
No. hav. (lat.
Lat. — Dec. :
65 44
(10)
— dec.) :
0.29451 (11)
Z:
75 34
(14)
No. hav. Z
0.37533 (13)
Comp'd alt. :
14 26
(15)
Obs'd alt. :
14 30
(16)
Diff.:
4
(17)
Index No. :
5872
Azimuth :
217°
Lat. diff. :
3'.2
Dep. :
2.4
Long. diff. :
3'.2
D. R. lat. :
42° 20' N.
(9)
D. R. long. :
35° 16' W. (3)
Sumner pt. lat. :
42 17 N.
(18)
Sumner pt. long. :
35 19 W. (19)
Azimuth of Sumner line : 307°
(20)
1 See footnote, p. 116.
120 . NAVIGATION
When the object observed is a star (cf. p. 104) or planetj
the choice between formulas (1) and (2), page 113, is not quite
the same as in the case of a solar observation. We must
use formula (1) if the star was on the east side of the sky
when observed, which might be called a "forenoon" observa-
tion of the star ; and we must use (2) if the star was on the
west side of the sky, giving an "afternoon" star observa-
tion. The use of the remaining formulas (3) to (6) is the
same as for the sun ; but T is now no longer the ship's appar-
ent time. Instead, it is the star's hour-angle (p. 104) ;
to find it for use in formulas (1) and (2), and in Table 11,
we must first calculate (p. 85) the Greenwich sidereal
time corresponding to the G. M. T. of the observation, as
taken from the chronometer, duly corrected for error and
rate ; and then use the following formulas :
(7) Greenwich hour-angle = Greenwich sidereal time — right ascen-
sion of star,
.R. I T = Greenwich hour-angle + D. R. longitude, if east,
\ T = Greenwich hour-angle — D. R. longitude, if west.
As an application of the Sumner method to a star observa-
tion, let us take the observation of Sirius, Dec. 17, 1917,
off Cape Agulhas, already treated as a time-sight (p. 105).
From the preliminary calculations there given, we have :
Greenwich hour-angle, line (11), page 105 2* 6m 34* (1)
D. R. longitude (p. 105) is 20° 41' E., or by
Table 9 (p. 249) 1 22 44 E. (2)
By formula (8) above, we add (1) and (2),
giving T 3 29 18 (3)
The star bore west 1 (p. 105) so we choose formula (2)
(p. 113), and write:
cos lat. (p. 106, line 19), 35° 20' S. by D. R.
(see Table 4, p. 231) 9.91158 (4)
cos dec. (p. 106, line 12), - 16° 36' (Tab. 4, p. 212) 9.98151 (5)
hav. T, 3* 29m 18" (line 3, above) (see Table 10, p. 263) 9.28872 (6)
Adding (4) to (6) gives, by formula (2), page 1 13, hav.Z, 9.18181 l (7)
1 See p. 116, footnote.
* Sum diminished by 20 (see footnote, p. 102).
NEWER NAVIGATION METHODS 121
Next we choose formula (3), page 113, since latitude and
declination are both — . We have :
By formula (3), lat. - dec. = 35° 20' - 16° 36' = 18° 44' (8)
We now use formula (5), page 113. We have:
No. hav. 18° 44' (8) (see Table 10, p. 254) 0.02649 (9)
No. hav. X* (7) (see Table 10, p. 261) 0.15194 (10)
Adding (9) and (10) gives No. hav. Z 0.17843 (11)
And Z, corresponding to (11) is found from
Table 10, page 262 49° 59' (12)
Then, by formula (6), page 113,
computed alt. = 90° - Z (12), or 40° 1' (13)
This computed altitude (13) must be compared
with the observed altitude, fully corrected.
This was (p. 106, line 18) 40° 2' (14)
Difference between (13) and (14), in minutes, or dis-
tance of Sumner point from D. R. point in miles
(p. 113) 1 mile (15)
Next we find the star's azimuth from Table 11, page 287.
The top argument for, entering the table is T, line (3),
and it must be found in the "afternoon" lines, since the star
bore W. The argument for the left-hand column is the
declination, line (5). Under T (p. 287), and opposite
declination, we find (approximately) the tabular index num-
ber 7550. Then we find the computed altitude, 40° (13),
in the right-hand column of the table (p. 289), and follow
along its horizontal line until we again reach the index
number 7550. The nearest to 7550 is 7544; and under
this number, at the foot of the column, we find the two
"afternoon" azimuths 260° and 280°.
These two numbers are so nearly equal that there is un-
certainty in choosing between them. Had the observer
taken the star's bearing by compass at the time of observa-
tion (p. 115), the uncertainty would be removed. But
in the absence of this information, we must have recourse
to Table 12 (p. 290), the auxiliary azimuth table. Enter-
ing this table with the pair of arguments of the present
1 No. hav. here obtained from hav. without finding the angle X
(p. 117, footnote).
122 NAVIGATION
problem: viz. latitude 35°, declination 17°, we find the
auxiliary angle 31°. The computed altitude (13) being
40°, is greater than the auxiliary angle, and the latitude is — .
Therefore, by the instructions (p. 115), the azimuth is
not between 90° and 270°. We therefore choose 280° as
our final azimuth, since 260°, the other possible value, is in
the prohibited area between 90° and 270°.
The computed altitude (13) being less than the observed
altitude, this observation places the Sumner point 1 mile
(15) from the D. R. point, and bearing from it 280°, the same
as the star's azimuth (p. 113). The traverse table (p. 156)
gives, for distance 1 and course 280°, latitude 0.2, departure
1.0. The longitude difference, by Table 2 (p. 172), is 1'.2,
for the departure 1 .0. Therefore, since azimuth 280° indicates
on the compass card that the Sumner point is W. and N.
of the D. R. point, we have :
lat. of Sumner point = - 35° 20' (4) + 0'.2 = - 35° 20' (16)
long, of Sumner point = 20° 41' E. (2) - 1'.2 = 20° 40' E. (17)
The bearing of the Sumner line will be 90° greater than
the star's azimuth (p. Ill) ; so we have :
Bearing of Sumner line = 280° + 90° = 370° ; or,
dropping 360° = 10° (18)
The foregoing calculation of the Sumner point from a
star observation can of course also be put in condensed form.
In doing so, we have repeated certain numbers from page 107
without references in parentheses. But numbers taken
from the extended calculation just given have their reference
numbers attached.
This condensed form, like the others previously given, is
the form of calculation which would be used in actual
navigation. It is most important, in the interest of numeri-
cal accuracy, to make all calculations upon forms ; and no
numbers should be written on the forms without having an
adjoining statement as to the meaning of the numbers.
NEWER NAVIGATION METHODS
123
SUMNER LINE, CONDENSED FORM. STAR
Watch time : 16* 29"» 48*
C. - W. : - 1 23 50
Chr. time : 15 5 58
Chr. corr'n : - 2 28
Obs'd alt. : 40° 3'
G. M. T. : 15 3 30
Index : + 5
R. A. mean sun : 17 42 10
Table 6 : - 1
Corr'n, past noon : 2 28
Table 7 : - 5
Greenw'h sid. time : 8 48 8
Corr'd alt. : 40 2
R. A. of Sirius : 6 41 34
Greenw'h hour-angle : 2 6 34
D. R. long. : 1 22 44 E.
(2)
T: 3 29 18
(3)
T or (24* - T) » : 3* 29"» 18*
(3) hav. : 9.28872 (6)
Dec. : - 16° 36'
cos : 9.98151 (5)
D. R. lat. : - 35 20
cos: 9.91158 (4)
Sum of 3 = hav. X:
9.18181 (7)
. No. hav. X :
0.15194 (10)
Lat. - Dec. : 18° 44' (8) ;
No. hav. : 0.02649 (9)
Sum of 2 = No. hav. Z :
0.17843 (11)
Z:
49° 59' (12)
Computed alt. = 90° - Z :
40 1 (13)
Obs'd alt., corr'd :
40 2 (14)
Diff.:
1 (15)
Index No. : 7550
Azimuth : 280°
Lat. diff. : 0'.2 Dep. : 1.0
Long. diff. : 1'.2
Sumner pt. lat. : - 35° 20' (16)
; long. : 20° 40' E. (17)
Bearing of Sumner line : 10° (18)
We have now, in the foregoing examples, illustrated the
manner of determining a Sumner line completely by ascer-
taining the latitude and longitude of one point on the line
(the Sumner point), and the bearing of the line itself at that
point. It may be desired to draw the line on the chart,
which will always interest the navigator if he is near the
coast and has a large-scale chart. To draw it, we merely
locate the Sumner point on the chart by its latitude and longi-
1 See footnote, p. 116.
124 NAVIGATION
tude, and then draw the line through the point so that it
will make with the meridian an angle equal to the bearing
which has been computed for the line. The Sumner line
should be extended in both directions from the Sumner
point, for any convenient distance, in such a way that the
point will be near the middle of the line.
We can now gain a better understanding as to Sumner
navigation by comparing the results obtained in one of the
foregoing examples with the corresponding calculation of
the same example as a time-sight. Thus from the same ob-
servation (pp. 104, 119)
As A TiME-SlGHT As A StTMNER OBSERVATION
From D. R. latitude 42° 20' N. ; From D. R. latitude 42° 20' N. ;
D. R. longitude 35° 16' W., we D. R. longitude 35° 16' W., we
found the ship's longitude to be found the Sumner point to be
35° 24' W. in latitude 42° 17' ; longitude 35°
19' W. ; and azimuth of Sumner
line, 307°.
Starting with the same observed altitude, and the same
D. R. position of the ship, we get quite different results by
the two methods of calculation. The time-sight gives us
nothing but a longitude ; and it will be the correct ship's
longitude only if the D. R. latitude was also correct (p. 101).
Therefore the time-sight calculation leaves us with both
latitude and longitude still affected by possible errors in the
D. R. latitude.
On the other hand, the Sumner calculation gives us both
a latitude and a longitude, but neither belongs to the ship's
position. They both belong to the position of the Sumner
point, but they are free from the effects of any D. R. errors.
They fix the Sumner point only, but they fix it correctly.
Furthermore, our knowledge that the ship is somewhere
on the Sumner line is also a fact, free from error. So what
we learn from the Sumner method is sure ; what we get by
the older methods is all really D. R. information in some
NEWER NAVIGATION METHODS 125
degree. The Sumner method is independent of D. R., an
advantage of which the value cannot be estimated too highly.
Furthermore, it can be shown mathematically (cf. p. Ill)
that a single observation can never really do more than
determine a line on which the ship must be. Even a noon-
sight does no more than this ; for in determining the ship's
latitude, it really only makes known a horizontal line (the
ship's latitude parallel) on the chart. In other words, for
a noon-sight the Sumner line is horizontal, or has a bearing
of 90°. And it will always come out 90°, if a noon-sight is
worked as a Sumner observation.
But the principal purpose of our present comparison of
the two methods of calculation is to warn the navigator
against falling into the error of imagining the ship to be at
the Sumner point. The observation does no more than tell
us where the Sumner point is, and that the ship is somewhere
on the line ; so far as the observation is concerned, all points
on the line are equally likely to be the ship's true position.
Therefore it is misleading to call the Sumner point the ship's
"most probable position." Were it so, a second observation,
made later in the day, would give another "most probable
position" of the ship. We should then be naturally led to
take as the ship's final location a point midway between the
two "most probables," ascribing their divergence to possible
errors of observation. But the ship's real position we already
know (p. Ill) to be at the intersection of the two Sumner
lines resulting from the two observations. And this inter-
secting point may be many miles from both "most proba-
bles," and from the above-mentioned midpoint between
them.
Less than two observations cannot fix the ship's position
completely; when two have been made, a correct applica-
tion of the Sumner method requires that the intersection
point of two Sumner lines be determined by calculation.
But before explaining the method of doing this, we must
describe an excellent alternative way of making Sumner
126 NAVIGATION
calculations such as we have given in the above examples.
The results are the same results as before, but they are
obtained with less work, and quite without logarithms, by
means of special tables such as our Table 13 (p. 292),1 which
we shall call Kelvin's Sumner Line Table.
This table has a pair of arguments (p. 11), a and 6, a ap-
pearing at the heads of the tabular columns, and b in the
left-hand column of each page. Corresponding to these
two arguments, the table gives two angles, K and Q ; so that
whenever a and b are given we can find the corresponding
K and Q ; or, if a and K should be given, we can find the
corresponding 6 and Q.
In the Sumner problem we obtain, by preparatory calcu-
lation (cf. pp. 119, 123), the following data:
Declination of sun (or star) ; D. R. latitude ; D. R. longitude ;
T, the ship's apparent time of the observation for the sun, or the
hour-angle for a star ;
and we wish to get the computed altitude and the azimuth.
The principle on which Table 13 depends is that the D. R.
latitude and longitude being always somewhat uncertain,
we can, if we choose, change them by reasonable amounts
before beginning our calculations. The Sumner point will
then be determined by its distance and bearing from the
changed D. R. point, instead of the original D. R. point.
By this device the tabular calculation is much facilitated.
The use of the table is easy after a little practice, the work
being divided into a series of separate operations. In de-
scribing these operations we have used small subscript num-
bers, to distinguish the several arguments, etc. ; as, for in-
stance, in Operation 1 we use a\, b\, Ki.
1 These tables were first published by Lord Kelvin in 1876.
More extended ones were recently issued by Lieutenant de Aquino,
of the Brazilian Navy; and these were reprinted by the Hydro-
graphic Office, United States Navy, in 1917. Aquino also improved
Kelvin's method of using his table.
NEWER NAVIGATION METHODS 127
OPERATION 1, requiring no interpolation. Enter Table 13
with :
Arg. ai = declination, taken without regard to + or — sign, and cor-
rect to the nearest whole degree only ;
Arg. 61 = T, if T is between 0* and 6* ;
= 12* - T, if T is between 6* and 12* ;
= T - 12*, if T is between 12* and 18*;
= 24* - T, if T is between 18* and 24*;
and before use 61 must be turned into degrees with
Table 9 (p. 249). It need be correct to the nearest
degree only. This proceeding will make fei always
less than 90°.
Then take from the table the tabular angle KI, also correct
to the nearest degree only.
OPERATION 2, requiring simple interpolation. Enter the
table a second time with :
Arg. o» = the KI, obtained in Operation 1.
Then, under this a?, run down the ^C-column until you
find the declination (taken without regard to + or — sign) ;
so that, in other words, K2 = declination.
Take from the table the angle Q2, which stands next to
the declination Kz, and also the &2, which is in the left-hand
argument column, in the same horizontal line with the
declination K2 in the /f-column. It will rarely be possible
to find the declination (which must this time be exact to
the nearest minute) in the K-column; so that a simple
interpolation will be necessary in getting $2 and 62. An
example of this interpolation will be found on page 129 ; and,
as we shall see, it is practically the only numerical calculation
required in the whole problem. The Kelvin method is very
much shorter than it looks.
The angle Q2 is used in choosing the longitude of the
"changed D. R. point"; the latitude of that point will be
found in Operation 3. To utilize Q2 for a sun observation,
calculate the Greenwich apparent time (G. A. T.) of the
128 NAVIGATION
observation, as on page 102, line (8), and turn it into de-
grees with Table 9 (page 249). Then :
(1) W. long, of changed D. R. point = G. A. T. - Q2, if, in Oper-
ation 1, T was less than 6*;
(2) W. long, of changed D. R. point = G. A. T. - (180° - Q2) if,
in Operation 1, T was between 6* and 12*;
(3) W. long, of changed D. R. point = G. A. T. - (180° + Q2) if,
in Operation 1, T was between 12* and 18*;
(4) W. long, of changed D. R. point = G. A. T. - (360° - Q2) if,
in Operation 1, T was between 18* and 24*.
When the subtractions in these formulas cannot be made,
the G. A. T. may be increased by 360° ; and when the west
longitude comes out greater than 180°, subtract it from 360°,
and call it east longitude.
In the case of a star, we must use, in the above formulas,
the Greenwich hour-angle, instead of the G. A. T. See
page 105, line (11), for the method of obtaining it.
OPERATION 3, requiring no interpolation. Enter the table
a third time with :
Arg. o8 = Ki, again as obtained in Operation 1.
(5) Arg. bs = 90° - (b2 + changed D. R. lat.), if latitude and
declination are of opposite signs, one + and
one — ;
(6) Arg. fcs = (bt + changed D. R. lat.) - 90°, if T was between
90° and 270°;
(7) Arg. 6, = 90° - (62 - changed D. R. lat,), if latitude is less
than 62;
(8) Arg. &, = 90° + (&2 - changed D. R. lat.), if latitude is
greater than &».
In choosing among formulas (5) to (8), give them pre-
cedence in order ; do not use (7) or (8) if the conditions
stated for (5) or (6) are satisfied. And at this point, use
your privilege of choosing any reasonable changed D. R. lati-
tude for the ship ; and choose one that differs as little as pos-
sible from the original D. R. latitude, and that yet makes
63 a whole number of degrees. In this way, all further
NEWER NAVIGATION METHODS 129
interpolation is avoided. Having once chosen among the
formulas, the latitude is used without regard to + or —
signs.
To complete Operation 3, having entered the table with
the pair of arguments a3 and &3, take out the tabular K3
and Q3.
K3 is now the computed altitude, to be used (p. 113) in
locating the Sumner. point from the changed D. R. point;
and Qs is the sun's true azimuth, which will always come
from the table less than 90°. If the ship is in the northern
hemisphere, this azimuth must be counted from the north
point of the horizon if, in Operation 3, we used formulas (6)
or (7) ; or from the south point of the horizon, if we used
formulas (5) or (8). With the ship in the southern hemi-
sphere, interchange the north and south points of the horizon
in these directions. And in both hemispheres, the azimuth
will of course be counted toward the east or west, according
as the observation was a "forenoon" or "afternoon" one
(cf. p. 120).
We shall now use Table 13 for the example given on page
119 in condensed form. We have (p. 127) :
OPERATION 1.
a\ = dec. = 23°, p. 119, line (1), to the nearest degree;
&! = T = 2h 41-" 31', p. 119, line (4) = 40°, to the nearest
degree ; and, with ai and bi as arguments, Table 13 gives
(p. 298) : KI = 36°, to the nearest degree.
OPERATION 2.
02 = K! = 36°.
Kz = 23° 24', p. 119, line (1)
and, with 02 and K2, we must find Q2 and 62. Running down
the column headed a = 36° (p. 302), we find :
When K2 = 23° 5', Q2 = 39° 43', b2 = 29°,
When K2 = 23° 51', Q2 = 40° 0', b2 = 30°.
We wish to interpolate for K2 = 23° 24', which is 19'
down from 23° 5' toward 23° 51'. The whole distance from
130 NAVIGATION
23° 5' to 23° 51' is 46'. Therefore we must interpolate
down £f of the whole interval from Q2 = 39° 43' to Q2 =
40° 0'. The difference between these two Q2's is 17' ; there-
fore the final Q2, belonging to K2 = 23° 24', is 39° 43' +
^ X 17' = 39° 43' + 7' = 39° 50'. Similarly, the difference
between the two 62's being 60', the final value of 62, for
K2 = 23° 24', is 29° + if X 60' = 29° 25'. These two
little interpolations are practically all the calculation required
in the whole problem.
To find the longitude of the changed D. R. point from the
above Q2 = 39° 50', we take from page 102, line (8),
Greenwich apparent time of observation, 5* 2m 35*
which, by Table 9 (p. 249) is, 75° 39'
We now use formula (1), page 128, because T, in Opera-
tion 1, was less than 6A. We get :
W. long, of ch'd D. R. pt. = G. A. T. - Q, = 75° 39' - 39° 50'
= 35° 49' W.
OPERATION 3.
03 = Ki = 36°.
The D. R. latitude is + 42° 20' (p. 119, line (9)) ; and as
the declination is — , we choose formula (5), page 128.
This, without changing the D. R. latitude, would give 63 =
90°-(&2+D. R.lat.) =90° -(29° 25'+ 42° 20') = 90°- 71° 45';
but by choosing a changed D. R. latitude of 42° 35', we shall
make 63 a whole number of degrees. So we have :
63 = 90° - (62+ changed D. R. latitude) = 90° - (29° 25'
+ 42° 35') = 90° - 72° = 18°.
Now we enter the table with the arguments a3 = 36°, and
63 = 18°, and obtain, without interpolation (p. 302) :
K> = computed altitude = 14° 29',
Qt = sun's true azimuth = 37° 22'.
This azimuth must be counted from the south point of
the horizon, since we used formula (5) in Operation 3 ; and
NEWER NAVIGATION METHODS 131
as the observation was an afternoon one, the correct azi-
muth will be S. 37° 22' W. (cf. p. 19). Counted in the United
States Navy way, from the north toward the east, and so
around to 360°, the azimuth will be 217° 22'.
On page 119, we found : Computed altitude, 14° 26'; azi-
muth, 217°.
This computed altitude differs by 3' from the value just
found by Table 13. The difference is due to our having
changed the D. R. point.
From the changed D. R. point, in latitude 42° 35' N. ;
longitude 35° 49' W., we now calculate (see Condensed Form,
next page) the position of the Sumner point to be : latitude
42° 34' N. ; longitude 35° 50' W. The former position, as
obtained on page 119, was : latitude 42° 17' N. ; longitude
35° 19' W.
These two Sumner point positions should lie on the
same Sumner line if the method of Table 13 gives correct
results; and they will satisfy this test, if the bearing
of a line joining them agrees with the azimuth of the
Sumner line, which is 217° + 90° = 307°. From the two
Sumner point positions we have : latitude difference = 17' ;
longitude difference = 31'; departure (Table 2, p. 174)
= 23.0. The traverse table (p. 164) gives, for latitude 17,
departure 23.0, the distance 28, course 307°. The agree-
ment is perfect, and shows that the same Sumner line
passes through both points, though they are 28 miles
apart. This test also shows that the calculation may
indicate any point on the Sumner line as the Sumner point,
if the D. R. position of the ship is uncertain : and so
we again call attention to the error of taking the cal-
culated Sumner point as the ship's most probable position
(cf. p. 125).
We now, as usual, repeat the above calculation by Table 13,
in condensed form, and including the final determination
of the position of the Sumner point from the changed D. R.
point.
132 NAVIGATION
SUMNER LINE BY TABLE 13, CONDENSED FORM. SUN
[The following is taken from page 119.]
Decl., 4* :
- 23° 23'.7
Eq
. of time : + 3™
' 22».3
H. D. :
0.1
H.
D. :
1.2
Decl., 4*59»»:
-23 24
Eq
. time : + 3
21.1
Watch time :
2* 29*
58*
Obs'd alt. :
14° 19'
C. -W.:
2 27
8
Index :
+ 4
Chr. time :
4 57
6
Table 6 :
+ 12
Chr. corr'n :
+ 2
8
Table 7 :
-5
G. M. T. :
4 59
14
Corr'd alt. :
14 30
Eq. of time :
+ 3
21
D. R. lat. :
42° 20' N.
G. app. time :
5 2
35
D. R. long. :
35° 16' W.
D. R. long. :
2 21
4 W.
(3)
Ship's app. time,
T: 2 41
31
(4)
[The following is calculated with Table 13.]
OPERATION 1 OPERATION 2
ai = dec. =23° at = K i = 36°
bi = T = 2^ 41" 31»(4) Ki = dec. = 23° 24'
= 40° Table 13, Qt = 39° 50'
Table 13, Ki = 36° Table 13, bt = 29° 25'
Greenwich app. time = 5* 2»> 35* = 75° 39'
By page 128, form. (1), W. long, of changed D. R. pt. = G. A. T. - Q,
= 35° 49' W.
Lat. of changed D. R. pt. = 42° 35' N.
OPERATION 3
a. = Ki = 36°
bi = 90° - (6, + changed D. R. lat.) = 18°
Table 13, K* = comp'd alt. - 14° 29'
Table 13, Qt = azimuth of sun = 37° 22'
or, by U. S. Navy = 217° 22'
Azimuth of Sumner line = 217° 22' + 90°
= 307° 22'
Dist. of Sumner pt. from changed
D. R. pt. = corr'd obs'd alt. — comp'd alt. = 1' or 1 mile
Bearing of Sumner pt. from changed D. R. pt. = 217°,
since comp'd alt. is less than obs'd alt.
Dist. 1, on course 217°, gives lat. diff., 0'.8; dep., 0.6; long, diff., 0'.8
Lat. of Sumner pt. = lat. of ch'd D. R. pt. - lat. diff. = 42° 34' N.
Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. = 35° 50' W.
A practised navigator can make the above complete calcu-
lation in a few minutes, as there are no logs used ; and any
one can easily obtain the necessary practice at sea by simply
forming the habit of working his sights both as time-sights
and as Sumners. To illustrate the subject further, we now
give, in condensed form, the Star Example of p. 123, worked
by Table 13.
NEWER NAVIGATION METHODS 133
SUMNER LINE BY TABLE 13, CONDENSED FORM. STAR
[The following is taken from page 123.]
Watch time: 16* 29™ 48« Obs'd alt. : 40° 3'
C. - W. : - 1 23 50 Index : + 5
Chr. time : 15 5 58 Table 6 : - 1
Chr. corr'n : - 2 28 Table 7 : - 5
G. M. T. : 15 3 30 Corr'd obs'd alt. : 40 2
R. A. mean sun : 17 42 10
Corr'n, past noon : 2 28 Dec. of Sirius : — 16 36
Greenwich sid. time : 8 48 8 D. R. lat. : - 35 20
R. A. of Sirius : 6 41 34
Green, hour-angle : 2 6 34
D. R. long. : 1 22 44 E.
T: 3 29 18
[The following is calculated with Table 13.]
OPERATION 1 OPERATION 2
01 = dec. =17° oi = Ki = 49°
61 = T = 3* 29" 18* Kt = dec. = 16° 36'
= 52° Table 13, Q, - 51° 57'
Table 13, Ki = 49° Table 13, bt = 25° 49'
By page 128, form. (1),
W. long, of changed D. R. pt. = Green, hour-angle — Qz1
339° 41'
20° 19' E.
Lat. of changed D. R. pt. = - 35° 49'
OPERATION 3
at = Ki = 49°
By form. (8), page 128, 6. = 90° + (61 - changed D. R. lat.) = 80°
Table 13, Ki = comp'd alt. = 40° 15'
Table 13, Q» = az. of Sirius = N. 81° 25' W.
or, by U. S. Navy = 278° 35'
Az. of Sumner line = 368° 35', or 8° 35'
Dist. of Sumner pt. from changed
D. R. pt. = corr'd obs'd alt. — comp'd alt. = — 13' or 13 miles
Bearing of Sumner pt. from changed D. R. pt. = 99°,
since comp'd alt. is greater than obs'd alt.
Dist. 13, on course 99°, gives lat. diff ., 2'.0 ; dep., 12.8 ; long, diff ., 15'.9
Lat. of Sumner pt. = lat. of ch'd D. R. pt. + lat. diff. = - 35° 51'
Long, of Sumner pt. = long, of ch'd D. R. pt. + long. diff. = 20° 35' E.
To complete this part of our subject, it remains to show
how the position of the ship can be found at the intersec-
tion of two Sumner lines (pp. Ill, 125) resulting from
two different observations. Figure 18 explains the nature of
the problem ; and it is almost exactly the same figure and
1 Qz being larger than the Greenwich hour-angle, the latter was
increased by 360°, to make the subtraction possible (p. 128).
134
NAVIGATION
problem treated in Chapter V, when we discussed fixing a
ship's position by means of "bearings from the bow"
(p. 54).
The two Sumner lines in Fig. 18 are SL and S'L, passing
through the two Sumner points S and Sf, whose latitudes
and longitudes are known
by calculation from the
observed altitudes. The
bearings or azimuths of the
two Sumner lines from the
north are the two angles
NSL and N'S'L, which are
also known from the pre-
vious calculations. It is
now required to find the
latitude and longitude of
the intersection point L,
where the ship is situated.
The similarity of this
problem to the former one
/^ in Chapter V becomes plain,
FIG. ^.-Intersection of Sumner Lines. if WG imaSine a SeCOnd shiP
sailing from one Sumner
point to the other, as from S to S', and taking bearings
from her bow upon our ship, located at L. These bearings
will be the two angles S'SL and S"S'L. If the second
of these angles should happen to be just twice as big
as the first, the distance S'L between the two ships at
the time of the second bearing would be equal (p. 54) to
the distance SS' run by the imagined ship between the two
observations.
This would enable us to fix the position of the imagined
ship at S', if L were a lighthouse ashore. But if L is our
ship, and S' a Sumner point of known position, the same
observations of bow bearings would fix the position of our
ship at L. Nor is it necessary (or possible) to measure
NEWER NAVIGATION METHODS 135
such imaginary bearings, or read the patent log to get the
distance run by an imagined ship.
For the distance and bearing of the second Sumner point
from the first can be obtained from their known latitudes
and longitudes with the traverse table. Thus the line SS'
(marked "distance") and the bearing (or course) angle
NSS' become known. Furthermore, the "bow bearing" at
S is the angle S'SL, and it is equal to the difference NSL —
NSS'. We have just seen that NSS' is obtained from the
traverse table ; and NSL is the calculated azimuth of the
Sumner line through S. In a similar way we get the other
"bow bearing" S"S'L. If this were twice the first one, the
"required distance" S'L in the figure would be equal to the
known distance SS' between the two Sumner points. If
not, it can be easily shown mathematically that :
(1) Required distance = known distance X a factor,
(2) log factor = sin S'SL - sin (S"S'L - S'SL).
By these simple formulas the required distance S'L might
be found : and as we also know the latitude and longitude
of the Sumner point S', and the azimuth or bearing of S'L,
the traverse table will make known the latitude and longi-
tude of the ship at L. It is to be noted also that as we are
at liberty to call either of the Sumner points S', it is desirable
to call that one S' which has the larger "bow bearing,"
so that there will be no difficulty about subtracting S'SL
from S"S'L.
The factor of formula (2) above can practically always
be found in our Table 14, the Sumner Intersection Table,
without using logarithms. The pair of arguments of the
table are the smaller "bow bearing" and the larger "bow
bearing"; the tabular number is the factor of formula (1)
above, and will always give the distance of the intersection
point from that one of the two Sumner points for which
the bow bearing was the larger.
And it should not be forgotten that the Sumner line really
136 NAVIGATION
extends equally in both directions (p. 124) from the Sumner
point, whereas, in Fig. 18, we have extended it mainly
in the direction of the intersection point L. Now the cal-
culated azimuth of any Sumner line may be changed 180°
at will, because the bearings of the two ends of the line from
the Sumner point differ by 180°, and we may take the bear-
ing of the line to be the bearing of either end from the Sumner
point in the middle of the line. Figure 18 shows, however,
that for the purpose of the present problem we must choose
the bearing of that end of the line which is nearest the point
of intersection L; nor does the choice ever offer difficulty,
because the known D. R. position of the ship at L, when
compared with the known positions of the two Sumner
points, will always indicate whether L bears east or west
of either Sumner point, and also whether it bears north or
south. And the bearing of L once chosen, we can always
find either of the two bow bearings by this formula :
(3) Bow bearing = bearing of Sumner line minus bearing
of the second Sumner point S' from the first point S.
In using formula (3) it is allowable to increase the bear-
ings of the Sumner lines by 360°, when necessary to make
the subtractions possible, and if the formula brings out bow
bearings larger than 180°, subtract them from 360°, and
proceed as before.
It is also always desirable to draw a rough sketch for
every intersection problem occurring on shipboard so as to
guard against accidental large errors like 90° or 180° in ob-
taining the two bow bearings; and also to make sure that
the latitude and longitude of the intersection point L are
correctly computed with the traverse table.
The foregoing assumes that the ship did not move from
the point L between the two sextant observations from which
the two Sumner lines were calculated. This will rarely
be the case, because it is very desirable that the two observa-
tions, if they are both sun observations, be separated by
NEWER NAVIGATION METHODS 137
three or four hours, if possible. The condition of an unmov-
ing ship will occur only if she is a sailing vessel becalmed,
or a steamer at anchor ; or if the two observations are made
at nearly the same time upon two different heavenly bodies,
such as two stars.
High accuracy in the resulting "fix" (p. 53) of the ship
will then be attained, if the azimuths of the two stars differ
by about 90° at the time of observation. The same favor-
able condition will be secured if one of the observations is
made upon a star near upper transit (pp. 89, 96), in the
twilight just before sunrise or after sunset; and the other
observation, at nearly the same time, upon the sun, when
it is about 12° or 15° above the horizon.
But if the ship has traveled a considerable distance between
the two observations, it is necessary to allow for such travel
before calculating the intersection point. Suppose she has
gone a distance D, upon a course C, by D. R., between the
two observations. Then simply find from Tables 1 and 2
the difference of latitude and longitude corresponding to
distance D and course C • and apply them as corrections to
the latitude and longitude of the Sumner point belonging
to the first observation. Everything else, including the
bearing of the first Sumner line, remaining unchanged,
the calculation then proceeds by Table 14, just as if the
ship had not moved. The computed intersection point is
then the ship's position at the time, of the second sextant
observation.
We shall now work some intersection examples.
Suppose we have two Sumner lines, as shown in the rough
sketch, Fig. 19, taken on board a ship becalmed. The
two sextant observations give :
FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S'
lat.1
long.
bearing of Sumner line
42°34'N. 42° 50' N.
35° 50' W. 35° 36' W.
307° 93° (changed to 273°)
1 As found on page 132.
138
NAVIGATION
The rough sketch, Fig. 19, having been made, and the
two "bow bearings" marked with little circular arcs as
shown, we call that one of the two Sumner points S', which
has the larger bow bearing ; and, for the point S', we change
FIG. 19. — Rough Sketch of Sumner Intersection.
the bearing of the Sumner line from 93° to 180° + 93° =
273°, so as to count the bearing for that end of the line which
is toward the intersection point L (p. 136). The other
bearing, 307°, for the point S, is already correctly counted.
We now have, from the two Sumner point latitudes and
longitudes : latitude difference =^ 16' ; longitude difference =
14' ; departure (Table 2, p. 174, for middle latitude 43°) =
10.2 ; and, for latitude difference = 16, departure = 10.2,
we find (Table 1, p. 162), distance = 19, course = 32°. The
distance between the two Sumner points is therefore 19
miles, and the bearing of S' from S is 32°.
Now we apply formula -(3), page 136, and find :
Smaller bow bearing at S = 307° - 32° = 275°.
Larger bow bearing at S' = 273° - 32° = 241°.
Being larger than 180°, these must be subtracted from
360° (p. 136), giving :
Smaller bow bearing = 85°; Larger bow bearing = 119°.
Next we refer to Table 14, and find with the smaller
bearing 85°, and the larger 119° the factor 1.78 (p. 322).
NEWER NAVIGATION METHODS 139
According to formula (1), page 135, we then have:
Required distance LS' = distance SS' X factor
= 19 X 1.78 = 33.8 miles.
Therefore the position of the ship at L is distant 33.8
miles from AS', and she bears 273°. With this distance and
bearing or course angle, the traverse table (p. 154) gives :
latitude = 1.8, departure = 33.8. For the departure 33.8,
Table 2 gives, for the middle latitude 43° (p. 174), differ-
ence longitude = 46'.2. The bearing 273° showing that the
intersection point L is N. and W. of Sf, we have :
Latitude of ship at L = 42° 50' N. + 1'.8 = 42° 51'.8 N.
Longitude of ship at L = 35° 36' W. + 46'.2 = 36° 22' W.
As a second example take the following two Sumner lines,
as shown in the rough sketch, Fig. 20. The two sextant
observations give :
FOR ONE SUMNER POINT, S FOR THE OTHER POINT, S'
lat. : 14° 26' N. 15° 30' N.
. long. : 77° 8' W. 76° 22'. 5 W.
bearing of line : 53° 135°
And suppose the ship, in the interval
between the two sextant observations, has
traveled a distance D = 31 miles, on course
C = 205°. We must begin (p. 137) by
shifting the first Sumner point S a dis-
tance D, on the course C. For this course
and distance, we have (Table 1, p. 160) :
lat., 28M; dep., 13.1; diff. long., 13'.5 FIG. 20. — Rough
(Table 2, p. 168). Sketch^ Sumner
Therefore, the latitude and longitude of
the first Sumner point must be corrected (p. 137) as follows :
For the point S, lat. = 14° 26' N. - 28M = 13° 58' N.
long. = 77° 8' W. + 13'.5 = 77° 21'. 5 W.
Bearing (unchanged) = 53°.
We now have, for the two Sumner points : lat. diff., 92' ;
140 NAVIGATION
long, diff., 59' ; dep., 57.0 (p. 169) ; dist., 108 miles (p. 162) ;
bearing of Sf from S, 32°.
Now we have, by formula (3), page 136 :
Smaller bow bearing at S = 53° - 32° = 21°.
Larger bow bearing at S' = 135° - 32° = 103°.
Table 14 (p. 319) gives the factor 0.36 ; so that the ship at
L is distant from S' 108 X .36 = 38.9 miles, and bears 135°.
For this distance and bearing we have (Table 1, p. 166),
latitude = 27'.6; departure = 27.6; and longitude differ-
ence (Table 2, p. 168) = 28'.6. Finally, then, at the time
of the second sextant observation, the ship at L was in
latitude 15° 30' N. - 27'.6 = 15° 2'.4 N. ; and in longitude
76° 22'.5 W. - 28'.6 = 75° 54' W.
CHAPTER X
A NAVIGATOR'S DAY AT SEA
THE present chapter contains a number of examples by
means of which the reader can gain facility in the use of the
methods set forth in the preceding pages.
The steam yacht Nav is bound from New York to
Colon, and the captain plans to take his departure from
the Sandy Hook Lightship, on Dec. 18, 1917, as early as
possible in the morning.
The first bit of navigation, to be accomplished before the
yacht leaves her anchorage in the "Horseshoe," is to ascer-
tain by D. R. methods the proper course to steer from
Sandy Hook. A glance at the track chart of the north
Atlantic shows that she must go by way of Crooked Island
Passage, and the Windward Passage between Cuba and
Haiti. It is also apparent from the chart that the first land
to be sighted among the islands is Watlings Island, and that
the proper course should pass to the eastward of it.
The position of Sandy Hook Lightship l is lat. 40° 28' N. ;
long. 73° 50' W. Hinchinbroke Rock, at the southern end
of Watlings Island, is in lat. 23° 57' N. ; long. 74° 28' W.
But the course should be shaped for a point about 12 miles
east of Watlings Island, to be perfectly safe. The position
of such a point is (approximately) lat. 23° 57' N. ; long. 74°
15' W.2
1 There is an excellent list of latitudes and longitudes in Bow-
ditch's " Navigator."
2 The difference between this longitude and that of Hinchinbroke
Rock is 13' ; but 13' here corresponds to about 12 miles, on account
of Table 2.
141
142 NAVIGATION
ABSTRACT OF LOG. Steam Yacht Nav, Dec. 18, 1917
PATENT
Loo
COMPASS
COUKSB
TRUE
COURSE
7 : 02 A.M.
Took departure from Sandy
Hook Lightship
26.2
S.
188°
7:21
Sunrise, observed azimuth
31.0
S.
188°
8:00
41.0
S.
188°
9:00
57.2
S.
188°
9:36
9:42
Bow bearing, Barnegat ....
Altitude and azimuth
67.0
69.1
S.
S.
188°
188°
9:57
Beam bearing, Barnegat . . .
72.5
S.
188°
(fix, lat. 39° 45' N. ; long.
73° 59' W.)
10:00
10:07
Changed course
73.4
75.3
S.
S.^E.
188°
182°
11:00
88.7
S.JE.
182°
11:42
Ex-mer. obs'n lat. 39° 19';
D. R. long. 73° 58'
98.5
S.iE.
182°
12:00
102.6
*S* £ -•_« 1
S.£E.
182°
1 : 00 P.M.
117.7
S.JE.
182°
2:00
133.0
S.JE.
182°
3:00
149.0
S.iE.
182°
4:00
163.8
S.JE.
182°
4:12
Alt. and az., fix, lat. 38° 11' ;
long. 73° 54'
166.9
S.iE.
182°
5:00
182.0
S.iE.
182°
6:00
197.2
S.fE.
182|°
By the method of page 20, the course from Sandy Hook
Lightship should be 181°, and the distance is 990 miles.
These numbers, and all subsequent numbers in the present
chapter, should be verified by the reader.
The distance being quite large, it is well to check it by
the logarithmic method, page 33. The result by this method
is: course 181° 14', distance 991.7 miles.
The chart also shows that this course will carry the yacht
very near Barnegat Light, on the coast of New Jersey. The
position of this light is lat. 39° 46' N. ; long. 74° 6' W. The
captain decides that it will be well to plan passing this light
A NAVIGATOR'S DAY AT SEA 143
at about 5 miles' distance. The position of a point 5 miles
east of Barnegat Light is lat. 39° 46' N., long. 73° 59' W. The
course and distance to this point from Sandy Hook Ship
are 189° and 42.5 miles. This course is so nearly the same
as the course to Watlings Island that the captain decides
to steer the 189° course.
All this work must be complete before reaching Sandy
Hook, for the course from the lightship must be ready for
the quartermaster before the lightship is passed. And
there is still more preliminary work. For the courses cal-
culated above are true courses (p. 43) and the quarter-
master must have the compass course, so that he may be
able to steer the yacht. The method of calculating the
compass course from the true course is given on page 48 ; and
in applying it the captain must have his deviation tables
at hand. We shall assume that the tables printed on pages 48
and 49 were the ones furnished by the compass adjuster for
the present voyage.
An examination of the Atlantic track chart shows that in
the vicinity of Sandy Hook, the variation, V, is 10° W., or
— 10°. By formula (3) (p. 49), we then have, since the true
course T is 189° :
Magnetic course = M = T - V = 189° - (- 10°) = 199°.
The second deviation table (p. 49) shows that when the
magnetic course (or magnetic bearing of ship's head) is 199°,
the deviation, D, is + 18°. Then, with V = - 10°, D = 18°,
formula (1), page 45, gives :
Compass error = E = V + D = - 10° + 18° = + 8°.
And from formula (2), page 45 :
Compass course C = T - E = 189° - 8° = 181° ;
and so the yacht must be steered on a 181° compass course
for Barnegat. But the quartermaster is to steer by " points "
so that the course nearest the 181° course is due south. The
captain decides to have the yacht steered due south by
144 NAVIGATION
compass, and is prepared to give the quartermaster his
orders as soon as Sandy Hook Lightship shall be reached.
The foregoing preliminary work having been completed
the previous day, the anchor is tripped at the Horseshoe
about an hour before daylight on Dec. 18, the weather being
fine, sea smooth, and wind light from the northwest. The
lightship is reached and passed at 7 : 02 A.M., ship's time, civil
reckoning, the ship then taking her departure. At that
moment, the patent log is read, and found to register 26.2
miles. The quartermaster gets his orders to steer south;
and all the above facts are duly recorded in the log-book.
And at every hour thereafter, 8, 9, 10, etc., a similar record
must be made in the log-book.
The next event is sunrise, which occurs at 7 : 21, very
soon after leaving the lightship. The sun's compass bearing
can then be very conveniently observed, and will furnish
an excellent check on the compass adjuster. This observa-
tion was made at 7 : 21 A.M., ship's time, civil reckoning,
corresponding to 19* 21m, Dec. 17, ship's apparent time,
astronomic reckoning; and the sun's bearing or azimuth
was 113° by compass. This was entered in the log-book,
and at the same time the patent log was read, and found to
be 31.0 miles.
To check the deviation table, the procedure was then as
follows :
By patent log the yacht had proceeded from the light-
ship a distance of 31.0 — 26.2 = 4.8 miles, on a compass
course of 180°, or true course of 188°; by D. R., she had
therefore reached the position lat. 40° 23' N. ; long. 73° 51' W.
The sun's declination, from the almanac, is — 23° 23', and
the (approximate1) T (p. 100) is 19* 21m. The sun's true
azimuth is found from Table 11 to be 121° ; and in using the
table for this purpose take the altitude of the sun, for the
1 If there is any chance of this T being much in error, the cap-
tain's watch, by which the observation is timed, must be compared
with the chronometer. See p. 94.
A NAVIGATOR'S DAY AT SEA 145
moment of sunrise, to be 0°. The observed compass azi-
muth having been 113°, formula (2), page 45, gave E = T-C
= 121° - 113° = +8°. Then from formula (1), page 45,
j) = E -V = +8°- (- 10°) = + 18°. As expected, this
deviation agrees with the deviation table, which would
not be likely to go wrong so soon after the beginning of a
voyage.
At 8 A.M. the patent log read 41.0; and at 9 A.M., 57.2.
The course was still S. by compass, or 188°, true course.
At 9 : 24 Barnegat Light was sighted by the lookout, and
the mate was ordered to take bow-and-beam bearings (p. 55)
upon it.
At 9 : 36, the light bore 225° by compass, or 45° from the
bow ; patent log, 67.0.
At 9* 42m 28' by his watch the captain took the altitude
of the sun's lower limb with the sextant, and found it to
be 18° 51'. Index correction was + 3', and height of eye,
15 feet. C. - W. was 4ft 51m 50* ; and the chr. correction
by the rate card was 4*, slow. Patent log, 69.1. At 9 : 45
by the watch, the sun's azimuth was again observed with
pelorus, and found to be 137°, compass bearing. It was
intended to work a Sumner line from the altitude by Kelvin's
table; and the pelorus observation was made because the
sun's true azimuth always comes out as a by-product, when
Kelvin's table is used, and so it is just as well to have an-
other check on the deviation table. This is the peculiar
advantage of Kelvin's table... Without any additional cal-
culations, the compass is always checked up on the very
course the ship is steering. This is just what the good
navigator wants.
The observations could not be worked up at once, be-
cause the captain wished to see the result of the mate's
bow-and-beam bearings. At 9 : 57 by the watch, Barnegat
bore abeam, on the starboard hand, or 270° by compass, the
yacht being still on the 180° compass course. Patent log
now 72.5.
E
146 NAVIGATION
Between the bow-and-beam bearings the run by log was
72.5 — 67 = 5.5 miles. Therefore the yacht is now 5.5
miles from Barnegat Light, and the compass bearing of the
light is 270°. The compass error being + 8°, the true bear-
ing of the light is 278° ; and the bearing of the yacht from
the light is the former bearing reversed, or 278° — 180° = 98°,
true. From this comes an accurate and complete position
of the yacht. Barnegat Light is in lat. 39° 46' N. ; long. 74° 6'
W. The yacht, 5.5 miles away on the bearing 98°, must, by
traverse table, be in lat. 39° 45' N. ; long. 73° 59' W.
At 10 A.M., the log was 73.4, course 188°, true.
Now the captain prepared to shape a new course to be
followed from the Barnegat bow-and-beam bearing "fix" in
the above lat. 39° 45' N. ; long. 73° 59' W., at 9 : 57.
Allowing ten minutes to work up the new course, the
captain plans to change course at 10 : 07. At that time
the ship, on her course of 188°, will be (at 15-knot speed)
2'.5 S. and practically 0' W. of the Barnegat position. So
the course will be changed when the yacht is in lat. 39° 42' N. ;
long. 73° 59' W., at 10 : 07. The course and distance from
there to the point 12 miles east of Hinchinbroke Rock are :
distance, 945 miles ; course, 181°, true, or 173° by compass.
Therefore, by the table on page 52, the quartermaster gets
the new course S4E. by compass, at 10 : 07. This corre-
sponds to 174° by compass, or 182° true course; and at
10 : 07, when the course was changed, the patent log read
75.3.
Now the Sumner line, from the observation at 9* 42m 28*
by the watch, was worked by Kelvin's table ; and the result
was :
Sumner point is in lat. 39° 50' N. ; long. 73° 56' W. ; bearing of
Sumner line 237°.
It is necessary, as a check, to ascertain whether this Sum-
ner line passes through the position obtained for the ship
by the Barnegat bearings. Before doing this, the Sumner
point must be shifted by the method of page 137, to allow for
A NAVIGATOR'S DAY AT SEA 147
the motion of the yacht between 9 : 42, when the sextant
observation was made, and 9 : 57, when Barnegat bore
abeam. The difference is 15 minutes, and in that time the
ship moved south 3.4 miles by the patent log and an in-
significant distance west.
Therefore the corrected Sumner data are :
Sumner point is in lat. 39° 46'.6 N. ; long. 73° 56' W. ; bearing of
Sumner line 237°.
If everything fits, this Sumner line must pass through the
Barnegat "fix" of the yacht in lat. 39° 45' N. ; long. 73° 59'
W., because the yacht must have been somewhere on the
line.
The traverse table shows that the bearing of a line passing
the Sumner point and the yacht's position is 235°, differing
only 2° from the Sumner line bearing ; so this check is satis-
factory. But a better way to check this matter is to deter-
mine the yacht's position from the intersection of two lines,
one of which is the Sumner line, and the other the beam bear-
ing of Barnegat Light. This can be done by the method of
page 133. The data of the problem are :
Sumner point : lat. 39° 46'.6 N.
long. 73° 56' W.
Line bears 237°
Barnegat Light : lat. 39° 46' N.
long. 74° 6' W.
Line bears 98°
We shall call Barnegat Light Sr ; and then formula (3),
page 136, gives, for the two bow bearings :
At Sumner point, S, 237° - 266° = 29°.
At Barnegat, S', 98° - 266° = 168°.
For these two bearings, Table 14 gives the factor 0.74, and
the yacht is placed 6 miles from Barnegat, on the 98° bear-
ing. The bow-and-beam observations gave 5.5 miles, so
the check by the Sumner line is excellent.
It remains for the captain to utilize the azimuth observa-
148 NAVIGATION
tion made at 9 : 45. The bearing of the Sumner line was
237°, and therefore the sun's true azimuth was 147°. The
observed azimuth, by pelorus (p. 145), was 137°. The com-
pass error was therefore + 10°. The variation being - 10°,
the deviation by formula (1), page 45, is D = 10° - ( - 10°) =
+ 20°.
On page 143 we found that the deviation table made this
deviation + 18° ; so that the table appears to require a
correction of +2°. The captain decides not to correct
the table for the present, unless later azimuth observations
shall confirm it, especially as the sunrise observation showed
the adjuster's results to be correct. Azimuth observa-
tions made when the sun is high in the sky are not quite
as reliable as sunrise ones. Moreover, the observation was
made at 9 : 45, whereas the altitude observation, for which
the true azimuth was calculated with Kelvin's table, was
made at 9 : 42, so that the true azimuth must have been in
error by the sun's azimuth change in three minutes. This
could have been avoided by giving the mate orders to ob-
serve the azimuth at about the same moment when the
captain took the altitude. Or, the sun's azimuth change
in three minutes might be taken from the azimuth table, and
the computed true azimuth duly corrected.
At 11 the log read 88.7, and the course was S.|E. by com-
pass, or 182°, true.
At about 11 : 30, the weather showing signs of becoming
thick, no preparations were made for a noon-sight by the
method of page 86 ; and rather than take the risk of losing his
noon observation altogether, the captain took an ex-me-
ridian altitude at llft 42TO 0* by his watch; log was 98.5;
the sextant reading 26° 55' ; index + 3' ; height of eye 15
ft. ; C. — W. was now 4* 51m 42* ; and chronometer slow 4*.
The observation was worked by Kelvin's table, and gave
the Sumner point in lat. 39° 20' N. ; long. 73° 40' W. ; bearing
of Sumner line 86°. Figure 21 is a rough sketch of this Sumner
line. It is very nearly horizontal ; had the observation been
A NAVIGATOR'S DAY AT SEA
149
L
Ship's
Position
39°204
made at noon precisely, it would have been perfectly hori-
zontal.
It would now have been possible to move up the Sumner
line observed at 9 : 42, and obtain an intersection to fix the
position of the yacht.
But this did not seem 73°J4o'
necessary to the cap-
tain, because of the
beam bearing obtained
at Barnegat at 9 : 57,
which gave a good fix.
And the present
Sumner line being so
nearly horizontal, it is
not necessary to know
the longitude very ac-
curately to obtain an
exact latitude. The
longitude by D. R. is
sufficient, and it is 73° 58' W. The difference between
this longitude and that of the Sumner point (73° 40') is
18' ; and the ship at L (fig. 21) bears 180° + 86° = 266°
from the Sumner point. Table 2 gives the dep. 14.0 for
long. diff. 18', in lat. 39°. And for course 266°, dep. 14.0,
we find in Table 1, lat. diff. I'.O, so the yacht's latitude is 1'
less than that of the Sumner point, and is therefore 39° 19'.
This happens to be in exact accord with the D. R. latitude,
which was also 39° 19'. This was perfectly satisfactory,
and the captain decided to carry this Sumner line forward
for an intersection, in case he should obtain an observation
in the afternoon.
At 12, the patent log read 102.6, course S.fE., 182° true ;
D. R. lat. 39° 15' ; long. 73° 58' ; distance to Watlings Island
918 miles.
Had the yacht been on a course other than almost due
south, it would have been necessary to set the watch and the
FIG. 21. — Sumner Line from ex-Meridian
Observation.
150 NAVIGATION
cabin clock to ship's apparent time. In fact, some naviga-
tors set their watches to ship's apparent time before every
observation (p. 94) :
at 1, log read 117.7, misty,
at 2, log read 133.0, misty,
at 3, log read 149.0 misty,
at 4, log read 163.8, clearing.
At 4* I2m 18s by the watch, the weather having cleared,
the altitude of the sun was found to be 4° 38' ; index + 4' ;
eye 15 ft. ; C. — W. 4* 51m 50* ; chronometer slow 4* ; log
166.9. Sun's azimuth, observed by the mate at the same
time, came out 224° by compass.
This observation was worked for a Sumner line by the
Kelvin table, and gave :
Position of Sumner point lat. 38° 6' N. ; long. 73° 49' W. ; bearing
of line 145° ; azimuth of sun 235°.
The Sumner line obtained at 11* 42m 0' was brought up to
the time of the present observation by D. R. (p. 137), giving :
position of 11:42 Sumner
point, after moving it, lat.
38° 12' N.; long. 73° 43' W. ;
bearing of the line 86°.
Both lines were then
sketched, as shown in Fig.
22. The point S is the
FIG. 22. — Rough Sketch of Sumner (moved) Sumner point from
Line Intersection. ^ U:42 observation) S'
that from the 4 : 12 observation. The intersection point L is
the position of the ship at 4 : 12, and it came out (p. 134) :
lat. 38° 11' N. ; long. 73° 54' W. The position brought up
by D. R. from 11 :42 was : lat. 38° 11' ; long. 74° 1' ; so that
there has been an easterly set of the current, amounting to
7' of longitude in 4| hours. The sun's true azimuth at
4 : 12 was 235°, from the Kelvin table ; and the pelorus
observation gave 224°. The compass error was therefore
A NAVIGATOR'S DAY AT SEA 151
+ 11°. The variation being — 10°, the deviation must
be D = 11° - ( - 10° =) + 21°. The deviation table made
this deviation + 18°, so that table seems to require a correc-
tion of +3°. The pelorus observation of 9 : 45 gave a correc-
tion of -f- 2° for the deviation table ; and as this is now
apparently confirmed, the captain decides to examine the
chart again, before finally shaping course for the night, to
see if the yacht has not perhaps moved into a region where
the variation is different from the Sandy Hook variation so
far used.
At 5 the log read 182.0, course was still 182° true.
The captain now prepared to shape the course for the
night, and to change his course, if necessary, at 6 : 00. His
first step was to obtain the D. R. position at 6 : 00, starting
from the observed position at 4 : 12. This gave position at
6 : 00, by D. R. : lat. 37° 41' ; long. 73° 55'. The easterly
current l of about 2' per hour set the yacht farther east about
3' between 4 : 12 and 6 : 00. Therefore he took the D. R.
position at 6 : 00 to be lat. 37° 41' ; long. 73° 52'. The posi-
tion of the point of destination, 12 miles east of Watlings
Island, is still : lat. 23° 57' ; long. 74° 15'. The true course
and distance to that point from the yacht's 6 : 00 position is
therefore, by traverse table : course 181|° ; dist. 824 miles.
A further examination of the track chart shows that the
variation, which was — 10° at Sandy Hook, is now — 8°.
The compass error, from the last pelorus observation,
was + 11°. Consequently, by the pelorus observation, the
compass course for the night should be 181|° — 11° = 170^°,
or S.fE. (see the Table on p. 52). Furthermore, the
variation being now — 8° and the error + 11° makes the
deviation Z)=#-F= + ll°-(-8°) = + 19°. The com-
pass adjuster's deviation of + 18° is therefore vindicated,
and the compass course S.fE. can be set for the night.
At 6 the log read 197.2, course S.fE., or 182J° true.
1 Doubtless the Gulf Stream.
152 NAVIGATION
In conclusion, the captain of the Nav hopes he has been
able to make his imagined proceedings clear enough to help
the young navigator in planning his own first day's work at
sea. May it be the first of many happy and successful days.
And let him not forget, when attempting to verify the
various calculations and problems of the Nav, that every
observation in this book has been prepared by calculation,
and none is the result of actual sextant observing. Should
inconsistencies or errors be found by any young navigator, it
is hoped that he will make them known so that they may be
corrected, in case the Nav shall be required to make another
voyage in a second edition.
LIST OF TABLES
1. Traverse Table; explained on pages 10 and 19; and its
use in the Sumner method on pages 113, 135 154
2. Conversion of longitude difference and departure ; ex-
plained on page 16 168
3. Number logarithms ; explained on page 23 178
4. Trigonometric logarithms ; explained on page 31 196
5. Meridional parts ; explained on page 35 241
6. Sextant Correction Table ; explained on page 72 247
7. Dip correction ; explained on page 73 247
8. Conversion of hours and minutes into decimals of a day ;
explained on page 80 248
9. Conversion of degrees and minutes of longitude and hours
and minutes of time 249
10. Haversines ; explained on page 99 250
11. Azimuth Table ; explained on page 113 284
12. Auxiliary Azimuth Table; explained on page 115 290
13. Kelvin's Sumner Line Table ; explained on page 126 292
14. Sumner Intersection Table; explained on page 135 318
PUBLISHERS' NOTE
Table 3, Number Logarithms, has been reprinted from "The
Macmillan Logarithmic and Trigonometric Tables," New York,
1917.
153
154
Table 1. Traverse Table
1°
2°
i Pt. 3°
4°
5°
£ Pt. 6°
7°
(179°, 181°,
(178°, 182°
(177°, 183°,
(176°, 184°
(175°, 185"
(174°, 186°
(173°, 187°,
DlST
359°)
358°)
357°)
356°)
355°)
354°)
353°)
Lat.
Dep.
Lat.
Dep
Lat.
Dep.
Lat.
Dep
Lat.
Dep.
Lat.
Dep
Lat.
Dep.
1
1.0
0.0
1.0
0.0
1.0
0.1
1.0
0.1
1.0
0.1
1.0
0.1
1.0
0.1
2
2.0
0.0
2.0
0.1
2.0
0.1
2.0
0.1
2.0
0.2
2.0
0.2
2.0
0.2
3
3.0
0.1
3.0
0.1
3.0
0.2
3.0
0.2
3.0
0.3
3.0
0.3
3.0
0.4
4
4.0
0.1
4.0
0.1
4.0
0.2
4.0
0.3
4.0
0.3
4.0
0.4
4.0
0.5
5
5.0
0.1
5.0
0.2
5.0
0.3
5.0
0.3
5.0
0.4
5.0
0.5
5.0
0.6
6
6.0
0.1
6.0
0.2
6.0
0.3
6.0
0.4
6.0
0.5
6.0
0.6
6.0
0.7
7
7.0
0.1
7.0
0.2
7.0
0.4
7.0
0.5
7.0
0.6
7.0
0.7
6.9
0.9
8
8.0
0.1
8.0
0.3
8.0
0.4
8.0
0.6
8.0
0.7
8.0
0.8
7.9
1.0
9
9.0
0.2
9.0
0.3
9.0
0.5
9.0
0.6
9.0
0.8
9.0
0.9
8.9
1.1
10
10.0
0.2
10.0
0.3
10.0
0.5
10.0
0.7
10.0
0.9
9.9
1.0
9.9
1.2
11
11.0
0.2
11.0
0.4
11.0
0.6
11.0
0.8
11.0
1.0
10.9
1.1
10.9
1.3
12
12.0
0.2
12.0
0.4
12.0
0.6
12.0
0.8
12.0
1.0
11.9
1.3
11.9
1.5
13
13.0
0.2
13.0
0.5
13.0
0.7
13.0
0.9
13.0
1.1
12.9
1.4
12.9
1.6
14
14.0
0.2
14.0
0.5
14.0
0.7
14.0
1.0
13.9
1.2
13.9
1.5
13.9
1.7
15
15.0
0.3
15.0
0.5
15.0
0.8
15.0
1.0
14.9
1.3
14.9
1.6
14.9
1.8
16
16.0
0.3
16.0
0.6
16.0
0.8
16.0
1.1
15.9
1.4
15.9
1.7
15.9
1.9
17
17.0
0.3
17.0
0.6
17.0
0.9
17.0
1.2
16.9
1.5
16.9
1.8
16.9
2.1
18
18.0
0.3
18.0
0.6
18.0
0.9
18.0
1.3
17.9
1.6
17.9
1.9
17.9
2.2
19
19.0
0.3
19.0
0.7
19.0
1.0
19.0
1.3
18.9
1.7
18.9
2.0
18.9
2.3
20
20.0
0.3
20.0
0.7
20.0
1.0
20.0
1.4
19.9
1.7
19.9
2.1
19.9
2.4
21
21.0
0.*
21.0
0.7
21.0
1.1
20.9
1.5
20.9
1.8
20.9
2.2
20.8
2.6
22
22.0
0.4
22.0
0.8
22.0
1.2
21.9
1.5
21.9
1.9
21.9
2.3
21.8
2.7
23
23.0
0.4
23.0
0.8
23.0
1.2
22.9
1.6
22.9
2.0
22.9
2.4
22.8
2.8
24
24.0
0.4
24.0
0.8
24.0
1.3
23.9
1.7
23.9
2.1
23.9
2.5
23.8
2.9
25
25.0
0.4
25.0
0.9
25.0
1.3
24.9
1.7
24.9
2.2
24.9
2.6
24.8
3.0
26
26.0
0.5
26.0
0.9
26.0
1.4
25.9
1.8
25.9
2.3
25.9
2.7
25.8
3.2
27
27.0
0.5
27.0
0.9
27.0
1.4
26.9
1.9
26.9
2.4
26.9
2.8
26.8
3.3
28
28.0
0.5
28.0
1.0
28.0
1.5
27.9
2.0
27.9
2.4
27.8
2.9
27.8
3.4
29
29.0
0.5
29.0
1.0
29.0
1.5
28.9
2.0
28.9
2.5
28.8
3.0
28.8
3.5
30
30.0
0.5
30.0
1.0
30.0
1.6
29.9
2.1
29.9
2.6
29.8
3.1
29.8
3.7
31
31.0
0.5
31.0
1.1
31.0
1.6
30.9
2.2
30.9
2.7
30.8
3.2
30.8
3.8
32
32.0
0.6
32.0
1.1
32.0
1.7
31.9
2.2
31.9
2.8
31.8
3.3
31.8
3.9
33
33.0
0.6
33.0
1.2
33.0
1.7
32.9
2.3
32.9
2.9
32.8
3.4
32.8
4.0
34
34.0
0.6
34.0
1.2
34.0
1.8
33.9
2.4
33.9
3.0
33.8
3.6
33.7
4.1
35
35.0
0.6
35.0
1.2
35.0
1.8
34.9
2.4
34.9
3.1
34.8
3.7
34.7
4.3
36
36.0
0.6
36.0
1.3
36.0
1.9
35.9
2.5
35.9
3.1
35.8
3.8
35.7
4.4
37
37.0
0.6
37.0
1.3
36.9
1.9
36.9
2.6
36.9
3.2
36.8
3.9
36.7
4.5
38
38.0
0.7
38.0
1.3
37.9
2.0
37.9
2.7
37.9
3.3
37.8
4.0
37.7
4.6
39
39.0
0.7
39.0
1.4
38.9
2.0
38.9
2.7
38.9
3.4
38.8
4.1
38.7
4.8
40
40.0
0.7
40.0
1.4
39.9
2.1
39.9
2.8
39.8
3.5
39.8
4.2
39.7
4.9
41
41.0
0.7
41.0
1.4
40.9
2.1
40.9
2.9
40.8
3.6
40.8
4.3
40.7
5.0
42
42.0
0.7
42.0
1.5
41.9
2.2
41.9
2.9
41.8
3.7
41.8
4.4
41.7
5.1
43
43.0
0.8
43.0
1.5
42.9
2.3
42.9
3.0
42.8
3.7
42.8
4.5
42.7
5.2
44
44.0
0.8
44.0
1.5
43.9
2.3
43.9
3.1
43.8
3.8
43.8
4.6
43.7
5.4
45
45.0
0.8
45.0
1.6
44.9
2.4
44.9
3.1
44.8
3.9
44.8
4.7
44.7
5.5
46
46.0
0.8
46.0
1.6
45.9
2.4
45.9
3.2
45.8
4.0
45.7
4.8
45.7
5.6
47
47.0
0.8
47.0
1.6
46.9
2.5
46.9
3.3
46.8
4.1
46.7
4.9
46.6
5.7
48
48.0
0.8
48.0
1.7
47.9
2.5
47.9
3.3
47.8
4.2
47.7
5.0
47.6
5.8
49
49.0
0.9
49.0
1.7
48.9
2.6
48.9
3.4
48.8
4.3
48.7
5.1
48.6
6.0
50
50.0
0.9
50.0
1.7
49.9
2.6
49.9
3.5
49.8
4.4
49.7
5.2
49.6
6.1
100
100.0
1.7
99.9
3.5
99.9
5.2
99.8
7.0
99.6
8.7
99.5
10.5
99.3
12.2
200
200.0
3.5
199.9
7.0
199.7
10.5
199.5
14.0
199.2
17.4
198.9
20.9
198.5
24.4
300
300.0
5.2
299.8
10.5
299.6
15.7
299.3
20.9
298.9
26.1
298.4
31.4
297.8
36.6
400
399.9
7.0
399.8
13.9
399.4
20.9
399.0
27.9
398.5
34.9
397.8
41.8
397.0
48.7
500
499.9
8.8
499.7
17.4
499.3
26.2
498.8
34.8
498.1
43.6
497.3
52.3
496.3
61.0
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(91°, 269°,
(92°, 268°,
(93°, 267°,
(94°, 266°,
(95°, 265°,
(96°, 264°,
(97°, 263°,
271°)
272°)
273°)
274°)
275°)
276°)
277°)
89°
88°
7fPt.87°
86°
85°
7|Pt.84°
83°
Table 1. Traverse Table
155
1°
2°
| Pt. 3°
4°
5°
£ Pt. 6°
7°
(179°, 181°
(178°, 182°,
(177°, 183°,
(176°, 184°,
(175°, 185°,
(174°, 186°,
(173°, 187°,
DlBT
359°)
358°)
357°)
356°)
355°)
354°)
353°)
Lat.
Dep
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
51.0
0.9
51.0
1.8
50.9
2.7
50.9
3.6
50.8
4.4
50.7
5.3
50.6
6.2
52
52.0
0.9
52.0
1.8
51.9
2.7
51.9
3.6
51.8
4.5
51.7
5.4
51.6
6.3
53
53.0
0.9
53.0
1.8
52.9
2.8
52.9
3.7
52.8
4.6
52.7
5.5
52.6
6.5
54
54.0
0.9
54.0
1.9
53.9
2.8
53.9
3.8
53.8
4.7
53.7
5.6
53.6
6.6
55
55.0
1.0
55.0
1.9
54.9
2.9
54.9
3.8
54.8
4.8
54.7
5.7
54.6
6.7
56
56.0
1.0
56.0
2.0
55.9
2.9
55.9
3.9
55.8
4.9
55.7
5.9
55.6
6.8
57
57.0
1.0
57.0
2.0
56.9
3.0
56.9
4.0
56.8
5.0
56.7
6.0
56.6
6.9
58
58.0
1.0
58.0
2.0
57.9
3.0
57.9
4.0
57.8
5.1
57.7
6.1
57.6
7.1
59
59.0
1.0
59.0
2.1
58.9
3.1
58.9
4.1
58.8
5.1
58.7
6.2
58.6
7.2
60
60.0
1.0
60.0
2.1
59.9
3.1
59.9
4.2
59.8
5.2
59.7
6.3
59.6
7.3
61
61.0
1.1
61.0
2.1
60.9
3.2
60.9
4.3
60.8
5.3
60.7
6.4
60.5
7.4
62
62.0
1.1
62.0
2.2
61.9
3.2
61.8
4.3
61.8
5.4
61.7
6.5
61.5
7.6
63
63.0
1.1
63.0
2.2
62.9
3.3
62.8
4.4
62.8
5.5
62.7
6.6
62.5
7.7
64
64.0
1.1
64.0
2.2
63.9
3.3
63.8
4.5
63.8
5.6
63.6
6.7
63.5
7.8
65
65.0
1.1
65.0
2.3
64.9
3.4
64.8
4.5
64.8
5.7
64.6
6.8
64.5
7.9
66
66.0
1.2
66.0
2.3
65.9
3.5
65.8
4.6
65.7
5.8
65.6
6.9
65.5
8rO
67
67.0
1.2
67.0
2.3
66.9
3.5
66.8
4.7
66.7
5.8
66.6
7.0
66.5
8.2
68
68.0
1.2
68.0
2.4
67.9
3.6
67.8
4.7
67.7
5.9
67.6
7.1
67.5
8.3
69
69.0
1.2
69.0
2.4
68.9
3.6
68.8
4.8
68.7
6.0
68.6
7.2
68.5
8.4
70
70.0
1.2
70.0
2.4
69.9
3.7
69.8
4.9
69.7
6.1
69.6
7.3
69.5
8.5
71
71.0
1.2
71.0
2.5
70.9
3.7
70.8
5.0
70.7
6.2
70.6
7.4
70.5
8.7
72
72.0
1.3
72.0
2.5
71.9
3.8
71.8
5.0
71.7
6.3
71.6
7 5
71.5
8.8
73
73.0
1.3
73.0
2.5
72.9
3.8
72.8
5.1
72.7
6.4
72.6
7.6
72.5
8.9
74
74.0
1.3
74.0
2.6
73.9
3.9
73.8
5.2
73.7
6.4
73.6
7.7
73.4
9.0
75
75.0
1.3
75.0
2.6
74.9
3.9
74.8
5.2
74.7
6.5
74.6
7.8
74.4
9.1
76
76.0
1.3
76.0
2.7
75.9
4.0
75.8
5.3
75.7
6.6
75.6
7.9
75.4
9.3
77
77.0
1.3
77.0
2.7
76.9
4.0
76.8
5.4
76.7
6.7
76.6
8.0
76.4
9.4
78
78.0
1.4
78.0
2.7
77.9
4.1
77.8
5.4
77.7
6.8
77.6
8.2
77.4
9.5
79
79.0
1.4
79.0
2.8
78.9
4.1
78.8
5.5
78.7
6.9
78.6
8.3
78.4
9.6
80
80.0
1.4
80.0
2.8
79.9
4.2
79.8
5.6
79.7
7.0
79.6
8.4
79.4
9.7
81
81.0
1.4
81.0
2.8
80.9
4.2
80.8
5.7
80.7
7.1
80.6
8.5
80.4
9.9
82
82.0
1.4
82.0
2.9
81.9
4.3
81.8
5.7
81.7
7.1
81.6
8.6
81.4
10.0
83
83.0
1.4
82.9
2.9
82.9
4.3
82.8
5.8
82.7
7.2
82.5
8.7
82.4
10.1
84
84.0
1.5
83.9
2.9
83.9
4.4
83.8
5.9
83.7
7.3
83.5
8.8
83.4
10.2
85
85.0
1.5
84.9
3.0
84.9
4.4
84.8
5.9
84.7
7.4
84.5
8.9
84.4
10.4
86
86.0
1.5
85.9
3.0
85.9
4.5
85.8
6.0
85.7
7.5
85.5
9.0
85.4
10.5
87
87.0
1.5
86.9
3.0
86.9
4.6
86.8
6.1
86.7
7.6
86.5
9.1
86.4
10.6
88
88.0
1.5
87.9
3.1
87.9
4.6
87.8
6.1
87.7
7.7
87.5
9.2
87.3
10.7
89
89.0
1.6
88.9
3.1
88.9
4.7
88.8
6.2
88.7
7.8
88.5
9.3
88.3
10.8
90
90.0
1.6
89.9
3.1
89.9
4.7
89.8
6.3
89.7
7.8
89.5
9.4
89.3
11.0
91
91.0
1.6
90.9
3.2
90.9
4.8
90.8
6.3
90.7
7.9
90.5
9.5
90.3
11.1
92
92.0
1.6
91.9
3.2
91.9
4.8
91.8
6.4
91.6
8.0
91.5
9.6
91.3
11.2
93
93.0
1.6
92.9
3.2
92.9
4.9
92.8
6.5
92.6
8.1
92.5
9.7
92.3
11.3
94
94.0
1.6
93.9
3.3
93.9
4.9
93.8
6.6
93.6
8.2
93.5
9.8
93.3
11.5
95
95.0
1.7
94.9
3.3
94.9
5.0
94.8
6.6
94.6
8.3
94.5
9.9
94.3
11.6
96
96.0
1.7
95.9
3.4
95.9
5.0
95.8
6.7
95.6
8.4
95.5
10.0
95.3
11.7
97
97.0
1.7
96.9
3.4
96.9
5.1
96.8
6.8
96.6
8.5
96.5
10.1
96.3
11.8
98
98.0
1.7
97.9
3.4
97.9
5.1
97.8
6.8
97.6
8.5
97.5
10.2
97.3
11.9
99
99.0
1.7
98.9
3.5
98.9
5.2
98.8
6.9
98.6
8.6
98.5
10.3
98.3
12.1
100
100.0
1.7
99.9
3.5
99.9
5.2
99.8
7.0
99.6
8.7
99.5
10.5
99.3
12.2
600
599.9
10.5
599.6
20.9
599.2
31.4
598.6
41.9
597.7
52.3
596.7
62.7
595.5
73.1
700
699.8
12.2
699.5
24.4
699.0
36.6
698.2
48.8
697.2
61.0
696.1
73.2
694.9
85.3
800
799.8
14.0
799.5
27.9
798.9
41.9
798.0
55.8
796.9
69.7
795.6
83.6
794.1
97.5
900
899.7
15.7
899.3
31.4
898.6
47.1
897.6
62.8
896.4
78.4
895.0
94.1
893.3
109.6
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(91°, 269°,
(92°, 268°
(93°, 267°,
(94°, 266°,
(95°, 265°,
(96°, 264°,
(97°, 263°.
271°)
272°)
273°)
274°)
275°)
276°)
277°)
89°
88°
71 Pt. 87°
86°
85°
7i Pt. 84°
83°
156
Table 1. Traverse Table
f Pt. 8°
9°
10°
1 Pt. 11°
12°
13°
1 \ Pt, 14°
(172°, 188°,
(171°, 189°,
(170°, 190°,
(169°, 191°,
(168°, 192°,
(167°, 193°,
(166°, 194°,
DlST.
352°)
351°)
350°)
349°)
348°)
347°)
346°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
1.0
0.1
1.0
0.2
1.0
0.2
1.0
0.2
1.0
0.2
1.0
0.2
1.0
0.2
2
2.0
0.3
2.0
0.3
2.0
0.3
2.0
0.4
2.0
0.4
1.9
0.4
1.9
0.5
3
3.0
0.4
3.0
0.5
3.0
0.5
2.9
0.6
2.9
0.6
2.9
0.7
2.9
0.7
4
4.0
0.6
4.0
0.6
3.9
0.7
3.9
0.8
3.9
0.8
3.9
0.9
3.9
1.0
5
5.0
0.7
4.9
0.8
4.9
0.9
4.9
1.0
4.9
1.0
4.9
1.1
4.9
1.2
6
5.9
0.8
5.9
0.9
5.9
1.0
5.9
1.1
5.9
1.2
5.8
1.3
5.8
1.5
7
6.9
1.0
6.9
1.1
6.9
1.2
6.9
1.3
6.8
1.5
6.8
1.6
6.8
1.7
8
7.9
1.1
7.9
1.3
7.9
1.4
7.9
1.5
7.8
1.7
7.8
1.8
7.8
1.9
9
8.9
1.3
8.9
1.4
8.9
1.6
8.8
1.7
8.8
1.9
8.8
2.0
8.7
2.2
10
9.9
1.4
9.9
1.6
9.8
1.7
9.8
1.9
9.8
2.1
9.7
2.2
9.7
2.4
11
10.9
1.5
10.9
1.7
10.8
1.9
10.8
2.1
10.8
2.3
10.7
2.5
10.7
2.7
12
11.9
1.7
11.9
1.9
11.8
2.1
11.8
2.3
11.7
2.5
11.7
2.7
11.6
2.9
13
12.9
1.8
12.8
2.0
12.8
2.3
12.8
2.5
12.7
2.7
12.7
2.9
12.6
3.1
14
13.9
1.9
13.8
2.2
13.8
2.4
13.7
2.7
13.7
2.9
13.6
3.1
13.6
3.4
15
14.9
2.1
14.8
2.3
14.8
2.6
14.7
2.9
14.7
3.1
14.6
3.4
14.6
3.6
16
15.8
2.2
15.8
2.5
15.8
2".8
15.7
3.1
15.7
3.3
15.6
3.6
15.5
3.9
17
16.8
2.4
16.8
2.7
16.7
3.0
16.7
3.2
16.6
3.5
16.6
3.8
16.5
4.1
18
17.8
2.5
17.8
2.9
17.7
3.1
17.7
3.4
17.6
3.7
17.5
4.0
17.5
4.4
19
18.8
2.6
18.8
3.0
18.7
3.3
18.7
3.6
18.6
4.0
18.5
4.3
18.4
4.6
20
19.8
2.8
19.8
3.1
19.7
3.5
19.6
3.8
19.6
4.2
19.5
4.5
19.4
4.8
21
20.8
2.9
20.7
3.3
20.7
3.6
20.6
4.0
20.5
4.4
20.5
4.7
20.4
5.1
22
21.8
3.1
21.7
3.4
21.7
3.8
21.6
4.2
21.5
4.6
21.4
4.9
21.3
5.3
23
22.8
3.2
22.7
3.6
22.7
4.0
22.6
4.4
22.5
4.8
22.4
5.2
22.3
5.6
24
23.8
3.3
23.7
3.8
23.6
4.2
23.6
4.6
23.5
5.0
23.4
5.4
23.3
5.8
25
24.8
3.5
24.7
3.9
24.6
4.3
24.5
4.8
24.5
5.2
24.4
5.6
24.3
6.0
26
25.7
3.6
25.7
4.1
25.6
4.5
25.5
5.0
25.4
5.4
25.3
5.8
25.2
6.3
27
26.7
3.8
26.7
4.2
26.6
4.7
26.5
5.2
26.4
5.6
26.3
6.1
26.2
6.5
28
27.7
3.9
27.7
4.4
27.6
4.9
27.5
5.3
27.4
5.8
27.3
6.3
27.2
6.8
29
28.7
4.0
28.6
4.5
28.6
5.0
28.5
5.5
28.4
6.0
28.3
6.5
28.1
7.0
30
29.7
4.2
29.6
4.7
29.5
5.2
29.4
5.7
29.3
6.2
29.2
6.7
29.1
7.3
31
30.7
4.3
30.6
4.8
30.5
5.4
30.4
5.9
30.3
6.4
30.2
7.0
30.1
7.5
32
31.7
4.5
31.6
5.0
31.5
5.6
31.4
6.1
31.3
6.7
31.2
7.2
31.0
7.7
33
32.7
4.6
32.6
5.2
32.5
5.7
32.4
6.3
32.3
6.9
32.2
7.4
32.0
8.0
34
33.7
4.7
33.6
5.3
33.5
5.9
33.4
6.5
33.3
7.1
33.1
7.6
33.0
8.2
35
34.7
4.9
34.6
5.5
34.5
6.1
34.4
6.7
34.2
7.3
34.1
7.9
34.0
8.5
36
35.6
5.0
35.6
5.6
35.5
6.3
35.3
6.9
35.2
7.5
35.1
8.1
34.9
8.7
37
36.6
5.1
36.5
5.8
36.4
6.4
36.3
7.1
36.2
7.7
36.1
8.3
35.9
9.0
38
37.6
5.3
37.5
5.9
37.4
6.6
37.3
7.3
37.2
7.9
37.0
8.5
36.9
9.2
39
38.6
5.4
38.5
6.1
38.4
6.8
38.3
7.4
38.1
8.1
38.0
8.8
37.8
9.4
40
39.6
6.6
39.5
6.3
39.4
6.9
39.3
7.6
39.1
8.3
39.0
9.0
38.8
9.7
41
40.6
5.7
40.5
6.4
40.4
7.1
40.2
7.8
40.1
8.5
39.9
9.2
39.8
9.9
42
41.6
5.8
41.5
6.6
41.4
7.3
41.2
8.0
41.1
8.7
40.9
9.4
40.8
10.2
43
42.6
6.0
42.5
6.7
42.3
7.5
42.2
8.2
42.1
8.9
41.9
9.7
41.7
10.4
44
43.6
6.1
43.5
6.9
43.3
7.6
43.2
8.4
43.0
9.1
42.9
9.9
42.7
10.6
45
44.6
6.3
44.4
7.0
44.3
7.8
44.2
8.6
44.0
9.4
43.8
10.1
43.7
10.9
46
45.6
6.4
45.4
7.2
45.3
8.0
45.2
8.8
45.0
9.6
44.8
10.3
44.6
11.1
47
46.5
6.5
46.4
7.4
46.3
8.2
46.1
9.0
46.0
9.8
45.8
10.6
45.6
11.4
48
47.5
6.7
47.4
7.5
47.3
8.3
47.1
9.2
47.0
10.0
46.8
10.8
46.6
11.6
49
48.5
6.8
48.4
7.7
48.3
8.5
48.1
9.3
47.9
10.2
47.7
11.0
47.5
11.9
50
49.5
7.0
49.4
7.8
49.2
8.7
49.1
9.5
48.9
10.4
48.7
11.2
48.5
12.1
100
99.0
13.9
98.8
15.6
98.5
17.4
98.2
19.1
97.8
20.8
97.4
22.5
97.0
24.2
200
198.1
27.8
197.5
31.3
197.0
34.7
196.3
38.2
195.6
41.6
194.9
45.0
194.1
48.4
300
297.1
41.8
296.3
46.9
295.4
52.1
294.5
57.2
293.4
62.4
292.3
67.5
291.1
72.6
400
396.1
55.7
395.1
62.6
393.9
69.5
392.6
76.3
391.3
83.1
389.8
90.0
388.1
96.7
500
495.1
69.6
493.8
78.2
492.4
86.8
490.8
95.4
489.1
104.0
487.2
112.4
485.1
121.0
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(98°, 262°,
(99°, 261°,
(100°, 260°,
(101°, 259°,
(102°, 258°,
(103°, 257°,
(104°, 256°,
278°)
279°)
280°)
281°)
282°)
283°)
284°)
7J Pt. 82°
81°
80°
7 Pt. 79°
78°
77°
6 f Pt. 76°
The 1-Pt. or 11° Courses are : N. by E., N. by W., S. by E., S. by W.
Table 1. Traverse Table
157
f Pt. 8°
9°
10°
1 Pt. 11°
12°
13°
HPt. 14°
(172°, 188°,
(171°, 189°,
(170°, 190°,
(169°, 191°,
(168°, 192°,
(167°, 193°,
(166°, 194°,
DlST.
352°)
351°)
350°)
349°)
348°)
347°)
346°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
50.5
7.1
50.4
8.0
50.2
8.9
50.1
9.7
49.9
10.6
49.7
11.5
49.5
12.3
52
51.5
7.2
51.4
8.1
51.2
9.0
51.0
9.9
50.9
10.8
50.7
11.7
50.5
12.6
53
52.5
7.4
52.3
8.3
52.2
9.2
52.0
10.1
51.8
11.0
51.6
11.9
51.4
12.8
54
53.5
7.5
53.3
8.4
53.2
9.4
53.0
10.3
52.8
11.2
52.6
12.1
52.4
13.1
55
54.5
7.7
54.3
8.6
54.2
9.6
54.0
10.5
53.8
11.4
53.6
12.4
53.4
13.3
56
55.5
7.8
55.3
8.8
55.1
9.7
55.0
10.7
54.8
11.6
54.6
12.6
54.3
13.5
57
56.4
7.9
56.3
8.9
56.1
9.9
56.0
10.9
55.8
11.9
55.5
12.8
55.3
13.8
58
57.4
8.1
57.3
9.1
57.1
10.1
56.9
11.1
56.7
12.1
56.5
13.0
56.3
14.0
59
58.4
8.2
58.3
9.2
58.1
10.2
57.9
11.3
57.7
12.3
57.5
13.3
57.2
14.3
60
59.4
8.4
59.3
9.4
59.1
10.4
58.9
11.4
58.7
12.5
58.5
13.5
58.2
14.5
61
60.4
8.5
60.2
9.5
60.1
10.6
59.9
11.6
59.7
12.7
59.4
13.7
59.2
14.8
62
61.4
8.6
61.2
9.7
61.1
10.8
60.9
11.8
60.6
12.9
60.4
13.9
60.2
15.0
63
62.4
8.8
62.2
9.9
62.0
10.9
61.8
12.0
61.6
13.1
61.4
14.2
61.1
15.2
64
63.4
8.9
63.2
10.0
63.0
11.1
62.8
12.2
62.6
13.3
62.4
14.4
62.1
15.5
65
64.4
9.0
64.2
10.2
64.0
11.3
63.8
12.4
63.6
13.5
63.3
14.6
63.1
15.7
66
65.4
9.2
65.2
10.3
65.0
11.5
64.8
12.6
64.6
13.7
64.3
14.8
64.0
16.0
67
66.3
9 3
66.2
10.5
66.0
11.6
65.8
12.8
65.5
13.9
65.3
15.1
65.0
16.2
68
67.3
9.5
67.2
10.6
67.0
11.8
66.8
13.0
66.5
14.1
66.3
15.3
66.0
16.5
69
68.3
9.6
68.2
10.8
68.0
12.0
67.7
13.2
67.5
14.3
67.2
15.5
67.0
16.7
70
69.3
9.7
69.1
11.0
68.9
12.2
68.7
13.4
68.5
14.6
68.2
15.7
67.9
16.9
71
70.3
9.9
70.1
11.1
69.9
12.3
69.7
13.5
69.4
14.8
69.2
16.0
68.9
17.2
72
71.3
10.0
71.1
11.3
70.9
12.5
70.7
13.7
70.4
15.0
70.2
16.2
69.9
17.4
73
72.3
10.2
72.1
11.4
71.9
12.7
71.7
13.9
71.4
15.2
71.1
16.4
70.8
17.7
74
73.3
10.3
73.1
11.6
72.9
12.8
72.6
14.1
72.4
15.4
72.1
16.6
71.8
17.9
75
74.3
10.4
74.1
11.7
73.9
13.0
73.6
14.3
73.4
15.6
73.1
16.9
72.8
18.1
76
75.3
10.6
75.1
11.9
74.8
13.2
74.6
14.5
74.3
15.8
74.1
17.1
73.7
18.4
77
76.3
10.7
76.1
12.0
75.8
13.4
75.6
14.7
75.3
16.0
75.0
17.3
74.7
18.6
78
77.2
10.9
77.0
12.2
76.8
13.5
76.6
14.9
76.3
16.2
76.0
17.5
75.7
18.9
79
78.2
11.0
78.0
12.4
77.8
13.7
77.5
15.1
77.3
16.4
77.0
17.8
76.7
19.1
80
79.2
11.1
79.0
12.5
78.8
13.9
78.5
15.3
78.3
16.6
77.9
18.0
77.6
19.4
81
80.2
11.3
80.0
12.7
79.8
14.1
79.5
15.5
79.2
16.8
78.9
18.2
78.6
19.6
82
81.2
11.4
81.0
12.8
80.8
14.2
80.5
15.6
80.2
17.0
79.9
18.4
79.6
19.8
83
82.2
11.6
82.0
13.0
81.7
14.4
81.5
15.8
81.2
17.3
80.9
18.7
80.5
20.1
84
83.2
11.7
83.0
13.1
82.7
14.6
82.5
16.0
82.2
17.5
81.8
18.9
81.5
20.3
85
84.2
11.8
84.0
13.3
83.7
14.8
83.4
16.2
83.1
17.7
82.8
19.1
82.5
20.6
86
85.2
12.0
84.9
13.5
84.7
14.9
84.4
16.4
84.1
17.9
83.8
19.3
83.4
20.8
87
86.2
12.1
85.9
13.6
85.7
15.1
85.4
16.6
85.1
18.1
84.8
19.6
84.4
21.0
88
87.1
12.2
86.9
13.8
86.7
15.3
86.4
16.8
86.1
18.3
85.7
19.8
85.4
21.3
89
88.1
12.4
87.9
13.9
87.6
15.5
87.4
17.0
87.1
18.5
86.7
20.0
86.4
21.5
90
89.1
12.5
88.9
14.1
88.6
15.6
88.3
17.2
88.0
18.7
87.7
20.2
87.3
21.8
91
90.1
12.7
89.9
14.2
89.6
15.8
89.3
17.4
89.0
18.9
88.7
20.5
88.3
22.0
92
91.1
12.8
90.9
14.4
90.6
16.0
90.3
17.6
90.0
19.1
89.6
20.7
89.3
22.3
93
92.1
12.9
91.9
14.5
91.6
16.1
91.3
17.7
91.0
19.3
90.6
20.9
90.2
22.5
94
93.1
13.1
92.8
14.7
92.6
16.3
92.3
17.9
91.9
19.5
91.6
21.1
91.2
22.7
95
94.1
13.2
93.8
14.9
93.6
16.5
93.3
18.1
92.9
19.8
92.6
21.4
92.2
23.0
96
95.1
13.4
94.8
15.0
94.5
16.7
94.2
18.3
93.9
20.0
93.5
21.6
93.1
23.2
97
96.1
13.5
95.8
15.2
95.5
16.8
95.2
18.5
94.9
20.2
94.5
21.8
94.1
23.5
98
97.0
13.6
96.8
15.3
96.5
17.0
96.2
18.7
95.9
20.4
95.5
22.0
95.1
23.7
99
98.0
13.8
97.8
15.5
97.5
17.2
97.2
18.9
96.8
20.6
96.5
22.3
96.1
24.0
100
99.0
13.9
98.8
15.6
98.5
17.4
98.2
19.1
97.8
20.8
97.4
22.5
97.0
24.2
600
594.2
83.5
592. f
93.8
590.9
104.2
589.0
114.5
586.9
124.7
584.6
135.0
582.2
145.1
700
693.3
97.4
691.3
109.4
689.5
121.5
687.1
133.6
684.7
145.5
682.1
157.5
679.2
169.3
800
792.3
111.4
790.2
125.1
787.9
139.0
785.2
152.6
782.5
166.3
779.4
180.0
776.2
193.6
900
891.3
125.2
888.8
140.8
886.3
156.3
883.3
171.7
880.2
187.1
S70.S
202.4
873.2
217.7
Dep
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(98°, 262°,
(99°, 261°,
(100°, 260°,
(101°, 259°,
(102°, 258°,
(103°, 257°,
(104°, 256°,
278°)
279°)
280°)
281°)
282°)
283°)
284°)
1\ Pt. 82°
81°
80°
7 Pt. 79°
78°
77°
61 Pt. 76°
The 7-Pt. or 79° Courses are : E. by N., W. by N., E. by S., W. by S.
158
Table 1. Traverse Table
15°
16°
IJPt. 17°
18°
19°
1J Pt, 20°
(165°, 195°,
(164°, 196°,
(163°, 197°,
(162°, 198°,
(161°, 199°,
(160°, 200°,
DlST.
345°)
344°)
343°)
342°)
341°)
340°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
1.0
0.3
1.0
0.3
1.0
0.3
1.0
0.3
0.9
0.3
0.9
0.3
2
1 Q
0.5
1.9
0.6
1.9
0.6
1.9
0.6
1.9
0.7
1.9
07
3
2.9
0.8
2.9
0.8
2.9
0.9
2.9
0.9
2.8
1.0
2.8
1.0
4
3.9
1.0
3.8
1.1
3.8
1.2
3.8
1.2
3.8
1.3
3.8
1.4
5
4.8
1.3
4.8
1.4
4.8
1.5
4.8
1.5
4.7
1.6
4.7
1.7
6
5.8
1.6
5.8
1.7
5.7
1.8
5.7
1.9
5.7
2.0
5.6
2.1
7
6.8
1.8
6.7
1.9
6.7
2.0
6.7
2.2
6.6
2.3
6.6
2.4
8
7.7
2.1
7.7
2.2
7.7
2.3
7.6
2.5
7.6
2.6
7.5
2.7
9
8.7
2.3
8.7
2.5
8.6
2.6
8.6
2.8
8.5
2.9
8.5
3.1
10
9.7
2.6
9.6
2.8
9.6
2.9
9.5
3.1
9.5
3.3
9.4
3.4
11
10.6
2.8
10.6
3.0
10.5
3.2
10.5
3.4
10.4
3.6
10.3
3.8
12
11.6
3.1
11.5
3.3
11.5
3.5
11.4
3.7
11.3
3.9
11.3
4.1
13
12.6
3.4
12.5
3.6
12.4
3.8
12.4
4.0
12.3
4.2
12.2
4.4
14
13.5
3.6
13.5
3.9
13.4
4.1
13.3
4.3
13.2
4.6
13.2
4.8
15
14.5
3.9
14.4
4.1
14.3
4.4
14.3
4.6
14.2
4.9
14.1
5.1
16
15.5
4.1
15.4
4.4
15.3
4.7
15.2
4.9
15.1
5.2
15.0
5.5
17
16.4
4.4
16.3
4.7
16.3
5.0
16.2
5.3
16.1
5.5
16.0
5.8
18
17.4
4.7
17.3
5.0
17.2
5.3
17.1
5.6
17.0
5.9
16.9
6.2
19
18.4
4.9
18.3
5.2
18.2
5.6
18.1
5.9
18.0
6.2
17.9
6.5
20
19.3
5.2
19.2
5.5
19.1
5.8
19.0
6.2
18.9
6.5
18.8
6.8
21
20.3
5.4
20.2
5.8
20.1
6.1
20.0
6.5
19.9
6.8
19.7
7.2
22
21.3
5.7
21.1
6.1
21.0
6.4
20.9
6.8
20.8
7.2
20.7
7.5
23
22.2
6.0
22.1
6.3
22.0
6.7
21.9
7.1
21.7
7.5
21.6
7.9
24
23.2
6.2
23.1
6.6
23.0
7.0
22.8
7.4
22.7
7.8
22.6
8.2
25
24.1
6.5
24.0
6.9
23.9
7.3
23.8
7.7
23.6
8.1
23.5
8.6
26
25 1
6.7
25.0
7.2
24.9
7.6
?47
8.0
24.6
8.5
24.4
89
27
26.1
7.0
26.0
7.4
25.8
7.9
25.7
8.3
25.5
8.8
25.4
9.2
28
27.0
7.2
26.9
7.7
26.8
8.2
26.6
8.7
26.5
9.1
26.3
9.6
29
28.0
7.5
27.9
8.0
27.7
8.5
27.6
9.0
27.4
9.4
27.3
9.9
30
29.0
7.8
28.8
8.3
28.7
8.8
28.5
9.3
28.4
9.8
28.2
10.3
31
29.9
8.0
29.8
8.5
29.6
9.1
29.5
9.6
29.3
10.1
29.1
10.6
32
30.9
8.3
30.8
8.8
30.6
9.4
30.4
9.9
30.3
10.4
30.1
10.9
33
31.9
8.5
31.7
9.1
31.6
9.6
31.4
10.2
31.2
10.7
31.0
11.3
34
32.8
8.8
32.7
9.4
32.5
9.9
32.3
10.5
32.1
11.1
31.9
11.6
35
33.8
9.1
33.6
9.6
33.5
10.2
33.3
10.8
33.1
11.4
32.9
12.0
36
34.8
9.3
34.6
9.9
34.4
10.5
34.2
11.1
34.0
11.7
33.8
12.3
37
35.7
9.6
35.6
10.2
35.4
10.8
35.2
11.4
35.0
12.0
34.8
12.7
38
36.7
9.8
36.5
10.5
36.3
11.1
36.1
11.7
35.9
12.4
35.7
13.0
39
37.7
10.1
37.5
10.7
37.3
11.4
37.1
12.1
36.9
12.7
36.6
13.3
40
38.6
10.4
38.5
11.0
38.3
11.7
38.0
12.4
37.8
13.0
37.6
13.7
41
39.6
10.6
39.4
11.3
39.2
12.0
39.0
12.7
38.8
13.3
38.5
14.0
42
40.6
10.9
40.4
11.6
40.2
12.3
39.9
13.0
39.7
13.7
39.5
14.4
43
41.5
11.1
41.3
11.9
41.1
12.6
40.9
13.3
40.7
14.0
40.4
14.7
44
42.5
11.4
42.3
12.1
42.1
12.9
41.8
13.6
41.6
14.3
41.3
15.0
45
43.5
11.6
43.3
12.4
43.0
13.2
42.8
13.9
42.5
14.7
42.3
15.4
46
44.4
11.9
44.2
12.7
44.0
13.4
43.7
14.2
43.5
15.0
43.2
15.7
47
45.4
12.2
45.2
13.0
44.9
13.7
44.7
14.5
44.4
15.3
44.2
16.1
48
46.4
12.4
46.1
13.2
45.9
14.0
45.7
14.8
45.4
15.6
45.1
16.4
49
47.3
12.7
47.1
13.5
46.9
14.3
46.6
15.1
46.3
16.0
46.0
16.8
50
48.3
12.9
48.1
13.8
47.8
14.6
47.6
15.5
47.3
16.3
47.0
17.1
100
96.6
25.9
96.1
27.6
95.6
29.2
95.1
30.9
94.6
32.6
94.0
34.2
200
193.2
51.8
192.3
55.1
191.3
58.5
190.2
61.8
189.1
65.1
187.9
68.4
300
289.8
77.6
288.4
82.7
286.9
87.7
285.3
92.7
283.7
97.7
281.9
102.6
400
386.3
103.5
384.5
110.2
382.5
117.0
380.4
123.6
378.2
130.2
375.9
136.8
500
483.0
129.4
480.6
137.8
478.1
146.2
475.5
154.5
472.8
162.8
469.9
171.0
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(105°, 255°,
(106°, 254°,
(107°, 253°,
(108°, 252°,
(109°, 251°,
(110°, 250°,
285°)
286°)
287°)
288°)
289°)
290°)
75°
74°
6£ Pt, 73°
72°
71°
6i Pt. 70°
Table 1. Traverse Table
159
15°
16°
H Pt.l7°
18°
19°
If Pt. 20°
(165°, 195°,
(164°, 196°,
(163°, 197°,
(162°, 198°,
(161°, 199°,
(160°, 200°,
DlST.
345°)
344°)
343°)
342°)
341°)
340°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
49.3
13.2
49.0
14.1
48.8
14.9
48.5
15.8
48.2
16.6
47.9
17.4
52
50.2
13.5
50.0
14.3
49.7
15.2
49.5
16.1
49.2
16.9
48.9
17.8
53
51.2
13.7
50.9
14.6
50.7
15.5
50.4
16.4
50.1
17.3
49.8
18.1
54
52.2
14.0
51.9
14.9
51.6
15.8
51.4
16.7
51.1
17.6
50.7
18.5
55
53.1
14.2
52.9
15.2
52.6
16.1
52.3
17.0
52.0
17.9
51.7
18.8
56
54.1
14.5
53.8
15.4
53.6
16.4
53.3
17.3
52.9
18.2
52.6
19.2
57
55.1
14.8
54.8
15.7
54.5
16.7
54.2
17.6
53.9
18.6
53.6
19.5
58
56.0
15.0
55.8
16.0
55.5
17.0
55.2
17.9
54.8
18.9
54.5
19.8
50
57.0
15.3
56.7
16.3
56.4
17.2
56.1
18.2
55.8
19.2
55.4
20.2
60
58.0
15.5
57.7
16.5
57.4
17.5
57.1
18.5
56.7
19.5
56.4
20.5
61
58.9
15.8
58.6
16.8
58.3
17.8
58.0
18.9
57.7
19.9
57.3
20.9
62
59.9
16.0
59.6
17.1
59.3
18.1
59.0
19.2
58.6
20.2
58.3
21.2
63
60.9
16.3
60.6
17.4
60.2
18.4
59.9
19.5
59.6
20.5
59.2
21.5
64
61.8
16.6
61.5
17.6
61.2
18.7
60.9
19.8
60.5
20.8
60.1
21.9
65
62.8
16.8
62.5
17.9
62.2
19.0
61.8
20.1
61.5
21.2
61.1
22.2
66
63.8
17.1
63.4
18.2
63.1
19.3
62.8
20.4
62.4
21.5
62.0
22.6
67
64.7
17.3
64.4
18.5
64.1
19.6
63.7
20.7
63.3
21.8
63.0
22.9
68
65.7
17.6
65.4
18.7
65.0
19.9
64.7
21.0
64.3
22.1
63.9
23.3
69
66.6
17.9
66.3
19.0
66.0
20.2
65.6
21.3
65.2
22.5
64.8
23.6
70
67.6
18.1
67.3
19.3
66.9
20.5
66.6
21.6
66.2
22.8
65.8
23.9
71
68.6
18.4
68.2
19.6
67.9
20.8
67.5
21.9
67.1
23.1
66.7
24.3
72
69.5
18.6
69.2
19.8
68.9
21.1
68.5
22.2
68.1
23.4
67.7
24.6
73
70.5
18.9
70.2
20.1
69.8
21.3
69.4
22.6
69.0
23.8
68.6
25.0
74
71.5
19.2
71.1
20.4
70.8
21.6
70.4
22.9
70.0
24.1
69.5
25.3
75
72.4
19.4
72.1
20.7
71.7
21.9
71.3
23.2
70.9
24.4
70.5
25.7
76
73.4
19.7
73.1
20.9
72.7
22.2
72.3
23.5
71.9
24.7
71.4
26.0
77
74.4
19.9
74.0
21.2
73.6
22.5
73.2
23.8
72.8
25.1
72.4
26.3
78
75.3
20.2
75.0
21.5
74.6
22.8
74.2
24.1
73.8
25.4
73.3
26.7
79
76.3
20.4
75.9
21.8
75.5
23.1
75.1
24.4
74.7
25.7
74.2
27.0
80
77.3
20.7
76.9
22.1
76.5
23.4
76.1
24.7
75.6
26.0
75.2
27.4
81
78.2
21.0
77.9
22.3
77.5
23.7
77.0
25.0
76.6
26.4
76.1
27.7
82
79.2
21.2
78.8
22.6
78.4
24.0
78.0
25.3
77.5
26.7
77.1
28.0
83
80.2
21.5
79.8
22.9
79.4
24.3
78.9
25.6
78.5
27.0
78.0
28.4
84
81.1
21.7
80.7
23.2
80.3
24.6
79.9
26.0
79.4
27.3
78.9
28.7
85
82.1
22.0
81.7
23.4
81.3
24.9
80.8
26.3
80.4
27.7
79.9
29.1
86
83.1
22.3
82.7
23.7
82.2
25.1
81.8
26.6
81.3
28.0
80.8
29.4
87
84.0
22.5
83.6
24.0
83.2
25.4
82.7
26.9
82.3
28.3
81.8
29.8
88
85.0
22.8
84.6
24.3
84.2
25.7
83.7
27.2
83.2
28.7
82.7
30.1
80
86.0
23.0
85.6
24.5
85.1
26.0
84.6
27.5
84.2
29.0
83.6
30.4
90
86.9
23.3
86.5
24.8
86.1
26.3
85.6
27.8
85.1
29.3
84.6
30.8
01
87.9
23.6
87.5
25.1
87.0
26.6
86.5
28.1
86.0
29.6
85.5
31.1
92
88.9
23.8
88.4
25.4
88.0
26.9
87.5
28.4
87.0
30.0
86.5
31.5
93
89.8
24.1
89.4
25.6
SS.9
27.2
88.4
28.7
87.9
30.3
87.4
31.8
94
90.8
24.3
90.4
25.9
89.9
27.5
89.4
29.0
88.9
30.6
88.3
32.1
95
91.8
24.6
91.3
26.2
90.8
27.8
90.4
29.4
89.8
30.9
89.3
32.5
96
92.7
24.8
92.3
26.5
91.8
28.1
91.3
29.7
90.8
31.3
90.2
32.8
97
93.7
25.1
93.2
26.7
92.8
28.4
92.3
30.0
91.7
31.6
91.2
33.2
98
94.7
25.4
94.2
27.0
93.7
28.7
93.2
30.3
92.7
31.9
92.1
33.5
99
95.6
25.6
95.2
27.3
94.7
28.9
94.2
30.6
93.6
32.2
93.0
33.9
100
96.6
25.9
96.1
27.6
95.6
29.2
95.1
30.9
94.6
32.6
94.0
34.2
600
579.5
155.3
576.8
165.4
573.8
175.4
570.6
185.4
567.3
195.3
563.8
205.2
700
676.1
181.1
672.8
193.0
669.4
204.6
665.8
216.3
661.9
227.9
657.9
239.4
800
772.7
207.0
769.0
220.5
765.0
233.9
760.8
247.3
756.5
260.4
751.8
273.6
900
869.2
232.9
865.0
248.0
860.6
263.1
855.9
278.1
850.9
_".)!'. '.1
845.7
307.8
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(105°. 255°,
(106°, 254°,
(107°, 253°,
(108°, 252°,
(109°, 251°,
(110°, 250°,
285°)
286°)
287°)
288°)
289°)
290°)
75°
74°
6i Pt. 73°
72°
71°
70°
160
Table 1. Traverse Table
21°
22°
2 Ft. 23°
24°
2| Ft. 25°
26°
DlST.
(159°, 201°,
(158°, 202°,
(157°, 203°,
(156°, 204°,
(155°, 205°,
(154°, 206°,
339°)
338°)
337°)
336°)
886°)
334°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
0.9
0.4
0.9
0.4
0.9
0.4
0.9
0.4
0.9
0.4
0.9
0.4
2
1.9
0.7
1.9
0.7
1.8
0.8
1.8
0.8
1.8
0.8
1.8
0.9
3
2.8
1.1
2.8
1.1
2.8
1.2
2.7
1.2
2.7
1.3
2.7
1.3
4
3.7
1.4
3.7
1.5
3.7
1.6
3.7
1.6
3.6
1.7
3.6
1.8
5
4.7
1.8
4.6
1.9
4.6
2.0
4.6
2.0
4.5
2.1
4.5
2.2
6
5.6
2.2
5.6
2.2
5.5
2.3
5.5
2.4
5.4
2.5
5.4
2.6
7
6.5
2.5
6.5
2.6
6.4
2.7
6.4
2.8
6.3
3.0
6.3
3.1
8
75
2.9
7.4
3.0
74
3.1
7.3
3.3
7.3
3.4
7.2
35
9
8.4
3.2
8.3
3.4
8.3
3.5
8.2
3.7
8.2
3.8
8.1
3.9
10
9.3
3.6
9.3
3.7
9.2
3.9
9.1
4.1
9.1
4.2
9.0
4.4
11
10.3
3.9
10.2
4.1
10.1
4.3
10.0
4.5
10.0
4.6
9.9
4.8
12
11.2
4.3
11.1
4.5
11.0
4.7
11.0
4.9
10.9
5.1
10.8
5.3
13
12 1
4.7
12.1
4.9
12.0
5.1
11.9
5.3
11.8
5.5
11.7
57
14
13.1
5.0
13.0
5.2
12.9
5.5
12.8
5.7
12.7
5.9
12.6
6.1
15
14.0
5.4
13.9
5.6
13.8
5.9
13.7
6.1
13.6
6.3
13.5
6.6
16
14.9
5.7
14.8
6.0
14.7
6.3
14.6
6.5
14.5
6.8
14.4
7.0
17
15.9
6.1
15.8
6.4
15.6
6.6
15.5
6.9
15.4
7.2
15.3
7.5
18
16.8
6.5
16.7
6.7
16.6
7.0
16.4
7.3
16.3
7.6
16.2
7.9
19
17.7
6.8
17.6
7.1
17.5
7.4
17.4
7.7
17.2
8.0
17.1
8.3
20
18.7
7.2
18.5
7.5
18.4
7.8
18.3
8.1
18.1
8.5
18.0
8.8
21
19.6
7.5
19.5
7.9
19.3
8.2
19.2
8.5
19.0
8.9
18.9
9.2
22
20.5
7.9
20.4
8.2
20.3
8.6
20.1
8.9
19.9
9.3
19.8
9.6
23
21.5
8.2
21.3
8.6
21.2
9.0
21.0
9.4
20.8
9.7
20.7
10.1
24
22.4
8.6
22.3
9.0
22.1
9.4
21.9
9.8
21.8
10.1
21.6
10.5
25
23.3
9.0
23.2
9.4
23.0
9.8
22.8
10.2
22.7
10.6
22.5
11.0
26
24.3
9.3
24.1
9.7
23.9
10.2
23.8
10.6
23.6
11.0
23.4
11.4
27
25.2
9.7
25.0
10.1
24.9
10.5
24.7
11.0
24.5
11.4
24.3
11.8
28
26.1
10.0
26.0
10.5
25.8
10.9
25.6
11.4
25.4
11.8
25.2
12.3
29
27.1
10.4
26.9
10.9
26.7
11.3
26.5
11.8
26.3
12.3
26.1
12.7
30
28.0
10.8
27.8
11.2
27.6
11.7
27.4
12.2
27.2
12.7
27.0
13.2
31
28.9
11.1
28.7
11.6
28.5
12.1
28.3
12.6
28.1
13.1
27.9
13.6
32
29.9
11.5
29.7
12.0
29.5
12.5
29.2
13.0
29.0
13.5
28.8
14.0
33
30.8
11.8
30.6
12.4
30.4
12.9
30.1
13.4
29.9
13.9
29.7
14.5
34
31.7
12.2
31.5
12.7
31.3
13.3
31.1
13.8
30.8
14.4
30.6
14.9
35
32.7
12.5
32.5
13.1
32.2
13.7
32.0
14.2
31.7
14.8
31.5
15.3
36
33.6
12.9
33.4
13.5
33.1
14.1
32.9
14.6
32.6
15.2
32.4
15.8
37
34.5
13.3
34.3
13.9
34.1
14.5
33.8
15.0
33.5
15.6
33.3
16.2
38
35 5
13.6
35.2
14.2
35.0
14.8
34.7
15.5
34.4
16.1
34.2
167
39
36.4
14.0
36.2
14.6
35.9
15.2
35.6
15.9
35.3
16.5
35.1
17.1
40
37.3
14.3
37.1
15.0
36.8
15.6
36.5
16.3
36.3
16.9
36.0
17.5
41
38.3
14.7
38.0
15.4
37.7
16.0
37.5
16.7
37.2
17.3
36.9
18.0
42
39.2
15.1
38.9
15.7
38.7
16.4
38.4
17.1
38.1
17.7
37.7
18.4
43
40.1
15.4
39.9
16.1
39.6
16.8
39.3
17.5
39.0
18.2
38.6
18.8
44
41.1
15.8
40.8
16.5
40.5
17.2
40.2
17.9
39.9
18.6
39.5
19.3
45
42.0
16.1
41.7
16.9
41.4
17.6
41.1
18.3
40.8
19.(
40.4
19.7
46
42.9
16.5
42.7
17.2
42.3
18.0
42.0
18.7
41.7
19.4
41.3
20.2
47
43.9
16.8
43.6
17.6
43.3
18.4
42.9
19.1
42.6
19.9
42.2
20.6
48
44.8
17.2
44.5
18.0
44.2
18.8
43.9
19.5
43.5
20.3
43.1
21.0
49
45.7
17.6
45.4
18.4
45.1
19.1
44.8
19.9
44.4
20.7
44.0
21.5
50
46.7
17.9
46.4
18.7
46.0
19.5
45.7
20.3
45.3
21.1
44.9
21.9
100
93.4
35.8
92.7
37.5
92.1
39.1
91.4
40.7
90.6
42.3
89.9
43.8
200
186.7
71.7
185.4
74.9
184.1
78.1
182.7
81.3
181.3
84.5
179.8
87.7
300
280.1
107.5
278.2
112.4
276.2
117.2
274.1
122.0
271.9
126.S
269.6
131.5
400
373.4
143.4
370.9
149.8
368.2
156.3
365.4
162.7
362.5
169.0
359.5
175.4
500
466.8
179.2
463.6
187.3
460.2
195.4
456.8
203.4
453.1
211.3
449.4
219.2
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(111°, 249°,
(112°, 248,°
(113°. 247°,
(114°, 246°,
(115°, 245°,
(116°, 244°,
291°)
292°)
293°)
294°)
295°)
296°)
69°
G Ft. 68°
67°
66°
5f Ft. 65°
64°
The 2-Pt. or 23° Courses are : N.N.E., N.N.W., S.S.E., S.S.W.
Table 1. Traverse Table
161
21°
22°
2 Pt. 23°
24°
21Pt, 25°
26°
(159°, 201°,
(158°, 202°,
(157°, 203°,
(156°, 204°,
(155°, 205°,
(154°, 206°,
DOT.
339°)
338°)
337°)
336°)
335°)
334°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
47.6
18.3
47.3
19.1
46.9
19.9
46.6
20.7
46.2
21.6
45.8
22.4
52
48.5
18.6
48.2
19.5
47.9
20.3
47.5
21.2
47.1
22.0
46.7
22.8
53
49.5
19.0
49.1
19.9
48.8
20.7
48.4
21.6
48.0
22.4
47.6
23.2
54
50.4
19.4
50.1
20.2
49.7
21.1
49.3
22.0
48.9
22.8
48.5
23.7
55
51.3
19.7
51.0
20.6
50.6
21.5
50.2
22.4
49.8
23,2
49.4
24.1
56
52.3
20.1
51.9
21.0
51.5
21.9
51.2
22.8
50.8
23.7
50.3
24.5
57
53.2
20.4
52.8
21.4
52.5
22.3
52.1
23.2
51.7
24.J
51.2
25.0
58
54.1
20.8
53.8
21.7
53.4
22.7
53.0
23.6
52.6
24.5
52.1
25.4
59
55.1
21.1
54.7
22.1
54.3
23.1
53.9
24.0
53.5
24.9
53.0
25.9
60
56.0
21.5
55.6
22.5
55.2
23.4
54.8
244
54,4
25.4
53.9
26.3
61
56.9
21.9
56.6
22.9
56.2
23.8
55.7
24.8
55.3
25.8
54.8
26.7
62
57.9
22.2
57.5
23.2
57.1
24.2
56.6
25.2
56.2
26.2
55.7
27.2
63
58.8
22.6
58.4
23.6
58.0
24.6
57.6
25.6
57.1
26.6
56.6
27.6
64
59.7
22.9
59.3
24.0
58.9
25.0
58.5
26.0
58.0
27.0
57.5
28.1
65
60.7
23.3
60.3
24.3
59.8
25.4
59.4
26.4
58.9
27.5
58.4
28.5
66
61.6
23.7
61.2
24.7
60.8
25.8
60.3
26.8
59.8
27.9
59.3
28.9
67
62.5
24.0
62.1
25.1
61.7
26.2
61.2
27.3
60.7
28.3
60.2
29.4
68
63.5
24.4
63.0
25.5
62.6
26.6
62.1
27.7
61.6
28.7
61.1
29.8
69
64.4
24.7
64.0
25.8
63.5
27.0
63.0
28.1
62.5
29.2
62.0
30.2
70
65.4
25.1
64.9
26.2
64.4
27.4
63.9
28.5
63.4
29.6
62.9
30.7
71
66.3
25.4
65.8
26.6
65.4
27.7
64.9
28.9
64.3
30.0
63.8
31.1
72
67.2
25.8
66.8
27.0
66.3
28.1
65.8
29.3
65.3
30.4
64.7
31.6
73
68.2
26.2
67.7
27.3
67.2
28.5
66.7
29.7
66.2
30.9
65.6
32.0
74
69.1
26.5
68.6
27.7
68.1
28.9
67.6
30.1
67.1
31.3
66.5
32.4
75
70.0
26.9
69.5
28.1
69.0
29.3
68.5
30.5
68.0
31.7
67.4
32.9
76
71.0
27.2
70.5
28.5
70.0
29.7
69.4
30.9
68.9
32.1
68.3
33.3
77
71.9
27.6
71.4
28.8
70.9
30.1
70.3
31.3
69.8
32.5
69.2
33.8
78
72.8
28.0
72.3
29.2
71.8
30.5
71.3
31.7
70.7
33'.0
70fl
34.2
79
73.0
28.3
73.2
29.6
72.7
30.9
72.2
32.1
71.6
33.4
71.0
34.6
80
74.7
28.7
74.2
30.0
73.6
31.3
73.1
32.5
72.5
33.8
71.9
35.1
81
75.6
29.0
75.1
30.3
74.6
31.6
74.0
32.9
73.4
34.2
72.8
35.5
82
76.6
29.4
76.0
30.7
75.5
32.0
74.9
33.4
74.3
34.7
73.7
35.9
83
77.5
29.7
77.0
31.1
76.4
32.4
75.8
33.8
75.2
35.1
74.6
36.4
84
78.4
30.1
77.9
31.5
77.3
32.8
76.7
34.2
76.1
35.5
75.5
36.8
85
79.4
30.5
78.8
31.8
78.2
33.2
77.7
34.6
77.0
35.9
76.4
37.3
86
80.3
30.8
79.7
32.2
79.2
33.6
78.6
35.0
77.9
36.3
77.3
37.7
87
81.2
31.2
80.7
32.6
80.1
34.0
79.5
35.4
78.8
36.8
78.2
38.1
88
82.2
31.5
81.6
33.0
81.0
34.4
80.4
35.8
79.8
37.2
79.1
38.6
89
83.1
31.9
82.5
33.3
81.9
34.8
81.3
36.2
80.7
37.6
80.0
39.0
90
84.0
32.3
83.4
33.7
82.8
35.2
82.2
36.6
81.6
38.0
80.9
39.5
91
85.0
32.6
84.4
34.1
83.8
35.6
83.1
37.0
82.5
38.5
81.8
39.9
92
85.9
33.0
85.3
34.5
84.7
35.9
84.0
37.4
83.4
38.9
82.7
40.3
93
86.8
33.3
86.2
34.8
85.6
36.3
85.0
37.8
84.3
39.3
83.6
40.8
94
87.8
33.7
87.2
35.2
86.5
36.7
85.9
38.2
85.2
39.7
84.5
41.2
95
88.7
34.0
88.1
35.6
87.4
37.1
86.8
38.6
86.1
40.1
85.4
41.6
96
89.6
34.4
89.0
36.0
88.4
37.5
87.7
39.0
87.0
40.6
86.3
42.1
97
90.6
34.8
89.9
36.3
89.3
37.9
88.6
39.5
87.9
41.0
87.2
42.5
98
91.5
35.1
90.9
36.7
90.2
38.3
89.5
39.9
88.8
41.4
88.1
43.0
99
92.4
35.5
91.8
37.1
91.1
38.7
90.4
40.3
89.7
41.8
89.0
43.4
100
93.4
35.8
92.7
37.5
92.1
39.1
91.4
40.7
90.6
42.3
89.9
43.8
600
560.1
215.0
556.3
224.8
552.3
234.4
548.1
244.0
543.8
253.6
539.3
263.0
700
653.6
250.8
649.1
262.2
644.3
273.5
639.5
284.7
634.5
295.8
629.2
306.8
800
746.9
286.7
741.8
299.7
736.4
312.6
730.8
325.4
725.1
338.1
719.1
350.6
900
840.3
322.5
s:i 1 ..',
337.1
828.3
351.7
822.1
:;<;r,. i)
815.6
:;,so.:;
808.9
394.5
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lafc.
Dep.
Lat.
(111°, 249°,
(112°, 248°,
(113°, 247°,
(114°, 246°,
(115°, 245°,
(116°, 244°,
291°)
292°)
293°)
294°)
295°)
296°)
69°
6 Pt. 68°
67°
66°
5| Pt.65°
64°
The 6-Pt. or 68° Courses are : E.N.E., W.N.W., E.S.E., W.S.W.
162
Table 1. Traverse Table
27°
2| Pt. 28°
29°
30°
2 f Pt. 31°
32°
(153°, 207°,
(152°, 208°,
(151°, 209°,
(150°, 210°,
(149°, 211°,
(148°, 212°,
DIST.
333°)
332°)
331°)
330°)
329°)
328°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
0.9
0.5
0.9
0.5
0.9
0.5
0.9
0.5
0.9
0.5
0.8
0.5
2
1.8
0.9
1.8
0.9
1.7
1.0
1.7
1.0
1.7
1.0
1.7
1.1
3
2.7
1.4
2.6
1.4
2.6
1.5
2.6
1.5
2.6
1.5
2.5
1.6
4
3.6
1.8
3.5
1.9
3.5
1.9
3.5
2.0
3.4
2.1
3.4
2.1
5
4.5
2.3
4.4
2.3
4.4
2.4
4.3
2.5
4.3
2.6
4.2
2.6
6
5.3
2.7
5.3
2.8
5.2
2.9
5.2
3.0
5.1
3.1
5.1
3.2
7
6.2
3.2
6.2
3.3
6.1
3.4
6.1
3.5
6.0
3.6
5.9
3.7
8
7.1
3.6
7.1
3.8
7.0
3.9
6.9
4.0
6.9
4.1
6.8
4.2
9
8.0
4.1
7.9
4.2
7.9
4.4
7.8
4.5
7.7
4.6
7.6
4.8
10
8.9
4.5
8.8
4.7
8.7
4.8
8.7
5.0
8.6
5.2
8.5
5.3
11
9.8
5.0
9.7
5.2
9.6
5.3
9.5
5.5
9.4
5.7
9.3
5.8
12
10.7
5.4
10.6
5.6
10.5
5.8
10.4
6.0
10.3
6.2
10.2
6.4
13
11.6
5.9
11.5
6.1
11.4
6.3
11.3
6.5
11.1
6.7
11.0
6.9
14
12.5
6.4
12.4
6.6
12.2
6.8
12.1
7.0
12.0
7.2
11.9
7.4
15
13.4
6.8
13.2
7.0
13.1
7.3
13.0
7.5
12.9
7.7
12.7
7.9
16
14.3
. 7.3
14.1
7.5
14.0
7.8
13.9
8.0
13.7
8.2
13.6
8.5
17
15.1
7.7
15.0
8.0
14.9
8.2
14.7
8.5
14.6
8.8
14.4
9.0
18
16.0
8.2
15.9
8.5
15.7
8.7
15.6
9.0
15.4
9.3
15.3
9.5
19
16.9
8.6
16.8
8.9
16.6
9.2
16.5
9.5
16.3
9.8
16.1
10.1
20
17.8
9.1
17.7
9.4
17.5
9.7
17.3
10.0
17.1
10.3
17.0
10.6
21
18.7
9.5
18.5
9.9
18.4
10.2
18.2
10.5
18.0
10.8
17.8
11.1
22
19.6
10.0
19.4
10.3
19.2
10.7
19.1
11.0
18.9
11.3
18.7
11.7
23
20.5
10.4
20.3
10.8
20.1
11.2
19.9
11.5
19.7
11.8
19.5
12.2
24
21.4
10:9
21:2
11.3
21.0
11.6
20.8
12.0
20.6
12.4
20.4
12.7
25
22.3
11.3
22.1
11.7
21.9
12.1
21.7
12.5
21 A
12.9
21.2
13.2
26
23.2
11.8
23.0
12.2
22.7
12.6
22.5
13.0
22.3
13.4
22.0
13.8
27
24.1
12.3
23.8
12.7
23.6
13.1
23.4
13.5
23.1
13.9
22.9
14.3
28
24.9
12.7
24.7
13.1
24.5
13.6
24.2
14.0
24.0
14.4
23.7
14.8
29
25.8
13.2
25.6
13.6
25.4
^A1
25.1
14.5
24.9
14.9
24.6
15.4
30
26.7
13.6
26.5
14.1
26.0
15.0
25.7
15.5
25.4
15.9
31
27.6
14.1
27.4
14.6
27.1
15.0
26.8
15.5
26.6
16.0
26.3
16.4
32
28.5
14.5
28.3
15.0
28.0
15.5
27.7
16.0
27.4
16.5
27.1
17.0
33
29.4
15.0
29.1
15.5
28.9
16.0
28.6
16.5
28.3
17.0
28.0
17.5
34
30.3
15.4
30.0
16.0
29.7
16.5
29.4
17.0
29.1
17.5
28.8
18.0
35
31.2
15.9
30.9
16.4
30.6
17.0
30.3
17.5
30.0
18.0
29.7
18.5
36
32.1
16.3
31.8
16.9
31.5
17.5
31.2
18.0
30.9
18.5
30.5
19.1
37
33.0
16.8
32.7
17.4
32.4
17.9
32.0
18.5
31.7
19.1
31.4
19.6
38
33.9
17.3
33.6
17.8
33.2
18.4
32.9
19.0
32.6
19.6
32.2
20.1
39
34.7
17.7
34.4
18.3
34.1
18.9
33.8
19.5
33.4
20.1
33.1
20.7
40
35.6
18.2
35.3
18.8
35.0
19.4
34.6
20.0
34.3
20.6
33.9
21.2
41
36.5
18.6
36.2
19.2
35.9
19.9
35.5
20.5
35.1
21.1
34.8
21.7
42
37.4
19.1
37.1
19.7
36.7
20.4
36.4
21.0
36.0
21.6
35.6
22.3
43
38.3
19.5
38.0
20.2
37.6
20.8
37.2
21.5
36.9
22.1
36.5
22.8
44
39.2
20.0
38.8
20.7
38.5
21.3
38.1
22.0
37.7
22.7
37.3
23.3
45
40.1
20.4
39.7
21.1
39.4
21.8
39.0
22.5
38.6
23.?
38.2
23.8
46
41.0
20.9
40.6
21.6
40.2
22.3
39.8
23.0
39.4
23.7
39.0
24.4
47
41.9
21.3
41.5
22.1
41.1
22.8
40.7
23.5
40.3
24.2
39.9
24.9
48
42.8
21.8
42.4
22.5
42.0
23.3
41.6
24.0
41.1
24.7
40.7
25.4
49
43.7
22.2
43.3
23.0
42.9
23.8
42.4
24.5
42.0
25.2
41.6
26.0
50
44.6
22.7
44.1
23.5
43.7
24.2
43.3
25.0
42.9
25.8
42.4
26.5
100
89.1
45.4
88.3
46.9
87.5
48.5
86.6
50.0
85.7
51.5
84.8
53.0
200
178.2
90.8
176.6
93.9
174.9
97.0
173.2
100.0
171.4
103.0
169.6
106.0
300
267.3
136.2
264.9
140.8
262.4
145.4
259.8
150.0
257.1
154.5
254.4
159.0
400
356.4
181.6
353.1
187.8
349.8
193.9
346.4
200.0
342.9
206.0
339.2
211.9
500
445.5
227.0
441.5
234.7
t:57.:^
242.4
433.0
250.0
428.6
257.5
424.0
265.0
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(117°, 243°,
(118°, 242°,
(119°, 241°,
(120°, 240°,
(121°, 239°,
(122°, 238°,
297°)
298°)
299°)
300°)
301°)
302°)
63°
5£Pt. 62°
61°
60°
5| Pt. 59°
58°
Table 1. Traverse Table
163
27°
2i Pt. 28°
29°
30°
2J Pt. 31°
32°
(153°, 207°,
(152°, 208°,
(151°, 209°,
(150°, 210°,
(149°, 21 1°,
(148°, 212°,
DlST.
333°)
332°)
331°)
330°)
329°)
328°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
45.4
23.2
45.0
23.9
44.6
24.7
44.2
25.5
43.7
26.3
43.3
27.0
52
46.3
23.6
45.9
24.4
45.5
25.2
45.0
26.0
44.6
26.8
44.1
27.6
53
47.2
24.1
46.8
24.9
46.4
25.7
45.9
26.5
45.4
27.3
44.9
28.1
54
48.1
24'.5
47.7
25.4
47.2
26.2
46.8
27.0
46.3
27.8
45.8
28.6
55
49.0
25.0
48.6
25.8
48.1
26.7
47.6
27.5
47J
28.3
46.6
29.1
56
49.9
25.4
49.4
26.3
49.0
27.1
48.5
28.0
48.0
28.8
47.5
29.7
57
50.8
25.9
50.3
26.8
49.9
27.6
49.4
28.5
48.9
29.4
48.3
30.2
58
51.7
26.3
51.2
27.2
50.7
28.1
50.2
29.0
49.7
29.9
49.2
30.7
59
52.6
26.8
52.1
27.7
51.6
28.6
51.1
29.5
50.6
30.4
50.0
31.3
60
53.5
27.2
53.0
28.2
52.5
29.1
52.0
30.0
51.4
30.9
50.9
31.8
61
54.4
27.7
53.9
28.6
53.4
29.6
52.8
30.5
52.3
31.4
51.7
32.3
62
55.2
28.1
54.7
29.1
54.2
30.1
53.7
31.0
53.1
31.9
52.6
32.9
63
56.1
28.6
55.6
29.6
55.1
30.5
54.6
31.5
54.0
32.4
53.4
33.4
64
57.0
29.1
56.5
30.0
56.0
31.0
55.4
32.0
54.9
33.0
54.3
33.9
65
57.9
29.5
57.4
30.5
56.9
31.5
56.3
32.5
55.7
33.5
55.1
34.4
66
58.8
30.0
58.3
31.0
57.7
32.0
57.2
33.0
56.6
34.0
56.0
35.0
67
59.7
30.4
59.2
31.5
58.6
32.5
58.0
33.5
57.4
34.5
56.8
35.5
68
60.6
30.9
60.0
31.9
59.5
33.0
58.9
34.0
58.3
35.0
57.7
36.0
69
61.5
31.3
60.9
32.4
60.3
33.5
59.8
.34.5
59.1
35.5
58.5
36.6
70
62.4
31.8
61.8
32.9
61.2
33.9
60.6
35.0
60.0
36.1
59.4
37.1
71
63.3
32.2
62.7
33.3
62.1
34.4
61.5
35.5
60.9
36.6
60.2
37.6
72
64.2
32.7
63.6
33.8
63.0
34.9
62.4
36.0
61.7
37.1
61.1
38.2
73
65.0
33.1
64.5
34.3
63.8
35.4
63.2
36.5
62.6
37.6
61.9
38.7
74
65.9
33.6
65.3
34.7
64.7
35.9
64.1
37.0
63.4
38.1
62.8
39.2
75
66.8
34.0
66.2
35.2
65.6
36.4
65.0
37.5
64.3
38.6
63.6
39.7
76
67.7
34.5
67.1
35.7
66.5
36.8
65.8
38.0
65.1
39.1
64.5
40.3
77
68.6
35.0
68.0
36.1
67.3
37.3
66.7
38.5
66.0
39.7
65.3
40.8
78
69.5
35.4
68.9
36.6
68.2
37.8
67.5
39.0
66.9
40.2
66.1
41.3
79
70.4
35.9
69.8
37.1
69.1
38.3
68.4
39.5
67.7
40.7
67.0
41.9
80
71.3
36.3
70.6
37.6
70.0
38.8
69.3
40.0
68.6
41.2
67.8
42.4
81
72.2
36.8
71.5
38.0
70.8
39.3
70.1
40.5
69.4
41.7
68.7
42.9
82
73.1
37.2
72.4
38.5
71.7
39.8
71.0
41.0
70.3
42.2
69.5
43.5
83
74.0
37.7
73.3
39.0
72.6
40.2
71.9
41.5
71.1
42.7
70.4
44.0
84
74.8
38.1
74.2
39.4
73.5
40.7
Z2.7
42.0
72.0
43.3
71.2
44.5
85
75.7
38.6
75.1
39.9
74.3
41.2
73.6
42.5
72.9
43.8
72.1
45.0
86
76.6
39.0
75.9
40.4
75.2
41.7
74.5
43.0
73.7
44.3
72.9
45.6
87
77.5
39.5
76.8
40.8
76.1
42.2
75.3
43.5
74.6
44.8
73.8
46.1
88
78.4
40.0
77.7
41.3
77.0
42.7
76.2
44.0
75.4
45.3
74.6
46.6
89
79.3
40.4
78.6
41.8
77.8
43.1
77.1
44.5
76.3
45.8
75.5
47.2
90
80.2
40.9
79.5
42.3
78.7
43.6
77.9
45.0
77.1
46.4
76.3
47.7
91
81.1
41.3
80.3
42.7
79.6
44.1
78.8
45.5
78.0
46.9
77.2
48.2
92
82.0
41.8
81.2
43.2
80.5
44.6
79.7
46.0
78.9
47.4
78.0
48.8
93
82.9
42.2
82.1
43.7
81.3
45.1
80.5
46.5
79.7
47.9
78.9
49.3
94
83.8
42.7
83.0
44.1
82.2
45.6
81.4
47.0
80.6
48.4
79.7
49.8
95
84.6
43.1
83.9
44.6
83.1
46.1
82.3
47.5
81.4
48.9
80.6
50.3
96
85.5
43.6
84.8
45.1
84.0
46.5
83.1
48.0
82.3
49.4
81.4
50.9
97
86.4
44.0
85.6
45.5
84.8
47.0
84.0
48.5
83.1
50.0
82.3
51.4
98
87.3
44.5
86.5
46.0
85.7
47.5
84.9
49.0
84.0
50.5
83.1
51.9
99
88.2
44.9
87.4
46.5
86.6
48.0
85.7
49.5
84.9
51.0
84.0
52.5
100
89.1
45.4
88.3
46.9
87.5
48.5
86.6
50.0
85.7
51.5
84.8
53.0
600
5346
272.4
529.8
281.7
524.8
290.9
519.6
300.0
514.3
309.0
508.8
3180
700
623.7
317.8
618.0
328.6
612.2
339.4
606.1
350.0
600.1
360.4
593.6
371.0
800
712.9
363.2
706.3
375.6
699.7
387.9
692.8
400.0
(isn.s
412.0
678.4
423.9
900
801.9
408.5
794.5
422.5
787.0
436.3
779.3
450.0
771.4
403.4
763.2
476.8
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(117°, 243°,
(118°, 242°,
(119°, 241°.
(120°, 240°
(121°, 239°,
(122°, 238°,
297°)
298°)
299°)
300°)
301°)
302°)
"
63°
5i Pt. 62°
61°
60°
5i Pt. 59°
58°
164
Table 1. Traverse Table
33°
3 Pt. 34°
35°
36°
3J Pt. 37°
38°
(147°, 213°,
(146°, 214°,
(145°, 215°,
(144°, 216°,
(143°, 217°,
(142°, 218°,
DlST.
327°)
326°)
325°)
324°)
323°)
322°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
0.8
0.5
0.8
0.6
0.8
0.6
0.8
0.6
0.8
0.6
0.8
0.6
2
1.7
1.1
1.7
1.1
1.6
1.1
1.6
1.2
1.6
1.2
1.6
1.2
3
2.5
1.6
2.5
1.7
2.5
1.7
2.4
1.8
2.4
1.8
2.4
1.8
4
3.4
2.2
3.3
2.2
3.3
2.3
3.2
2.4
3.2
2.4
3.2
2.5
5
4.2
2.7
4.1
2.8
4.1
2.9
4.0
2.9
4.0
3.0
3.9
3.1
6
5.0
3.3
5.0
3.4
4.9
3.4
4.9
3.5
4.8
3.6
4.7
3.7
7
5.9
3.8
5.8
3.9
5.7
4.0
5.7
4.1
5.6
4.2
5.5
4.3
8
6.7
4.4
6.6
4.5
6.6
4.6
6.5
4.7
6.4
4.8
6.3
4.9
9
7.5
4.9
7.5
5.0
7.4
5.2
7.3
5.3
7.2
5.4
7.1
5.5
10
8.4
5.4
8.3
5.6
8.2
5.7
8.1
5.9
8.0
6.0
7.9
6.2
11
9.2
6.0
9.1
6.2
9.0
6.3
8.9
6.5
8.8
6.6
8.7
6.8
12
10.1
6.5
9.9
6.7
9.8
6.9
9.7
7.1
9.6
7.2
9.5
7.4
13
10 Q
7.1
10.8
7.3
10.6
7.5
10.5
7.6
10.4
7.8
10.2
80
14
11.7
7.6
11.6
7.8
11.5
8.0
11.3
8.2
11.2
8.4
11.0
8.6
15
12.6
8.2
12.4
8.4
12.3
8.6
12.1
8.8
12.0
9.0
11.8
9.2
16
13.4
8.7
13.3
8.9
13.1
9.2
12.9
9.4
12.8
9.6
12.6
9.9
17
14.3
9.3
14.1
9.5
13.9
9.8
13.8
10.0
13.6
10.2
13.4
10.5
18
15.1
9.8
14.9
10.1
14.7
10.3
14.6
10.6
14.4
10.8
14.2
11.1
19
15.9
10.3
15.8
10.6
15.6
10.9
15.4
11.2
15.2
11.4
15.0
11.7
20
16.8
10.9
16.6
11.2
16.4
11.5
16.2
11.8
16.0
12.0
15.8
12.3
21
17.6
11.4
17.4
11.7
17.2
12.0
17.0
12.3
16.8
12.6
16.5
12.9
22
18.5
12.0
18.2
12.3
18.0
12.6
17.8
12.9
17.6
13.2
17.3
13.5
23
19.3
12.5
19.1
12.9
18.8
13.2
18.6
13.5
18.4
13.8
18.1
14.2
24
20.1
13.1
19.9
13.4
19.7
13.8
19.4
14.1
19.2
14.4
18.9
14.8
25
21.0
13.6
20.7
14.0
20.5
14.3
20.2
14.7
20.0
15.0
19.7
15.4
26
21.8
14.2
21.6
14.5
21.3
14.9
21.0
15.3
20.8
15.6
20.5
16.0
27
22.6
14.7
22.4
15.1
22.1
15.5
21.8
15.9
21.6
16.2
21.3
16.6
28
23.5
15.2
23.2
15.7
22.9
16.1
22.7
16.5
22.4
16.9
22.1
17.2
29
24.3
15.8
24.0
16.2
23.8
16.6
23.5
17.0
23.2
17.5
22.9
17.9
30
25.2
16.3
24.9
16.8
24.6
17.2
24.3
17.6
24.0
18.1
23.6
18.5
31
26.0
16.9
25.7
17.3
25.4
17.8
25.1
18.2
24.8
18.7
24.4
19.1
32
26.8
17.4
26.5
17.9
26.2
18.4
25.9
18.8
25.6
19.3
25.2
19.7
33
27.7
18.0
27.4
18.5
27.0
18.9
26.7
19.4
26.4
19.9
26.0
20.3
34
28.5
18.5
28.2
19.0
27.9
19.5
27.5
20.0
27.2
20.5
26.8
20.9
35
29.4
19.1
29.0
19.6
28.7
20.1
28.3
20.6
28.0
21.1
27.6
21.5
36
30?
19.6
29.8
20.1
29.5
20.6
29.1
21.2
28.8
21.7
28.4
??,2
37
31.0
20.2
30.7
20.7
30.3
21.2
29.9
21.7
29.5
22.3
29.2
22.8
38
31.9
20.7
31.5
21.2
31.1
21.8
30.7
22.3
30.3
22.9
29.9
23.4
39
32.7
21.2
32.3
21.8
31.9
22.4
31.6
22.9
31.1
23.5
30.7
24.0
40
33.5
21.8
33.2
22.4
32.8
22.9
32.4
23.5
31.9
24.1
31.5
24.6
41
34.4
22.3
34.0
22.9
33.6
23.5
33.2
24.1
32.7
24.7
32.3
25.2
42
35.2
22.9
34.8
23.5
34.4
24.1
34.0
24.7
33.5
25.3
33.1
25.9
43
36.1
23.4
35.6
24.0
35.2
24.7
34.8
25.3
34.3
25.9
33.9
26.5
44
36.9
24.0
36.5
24.6
36.0
25.2
35.6
25.9
35.1
26.5
34.7
27.1
45
37.7
24.5
37.3
25.2
36.9
25.8
36.4
26.5
35.9
27.1
35.5
27.7
46
38.6
25.1
38.1
25.7
37.7
26.4
37.2
27.0
36.7
27.7
36.2
28.3
47
39.4
25.6
39.0
26.3
38.5
27.0
38.0
27.6
37.5
28.3
37.0
28.9
48
40.3
26.1
39.8
26.8
39.3
27.5
38.8
28.2
38.3
28.9
37.8
29.6
49
41.1
26.7
40.6
27.4
40.1
28.1
39.6
28.8
39.1
29.5
38.6
30.2
50
41.9
27.2
41.5
28.0
41.0
28.7
40.5
29.4
39.9
30.1
39.4
30.8
100
83.9
54.5
82.9
55.9
81.9
57.4
80.9
58.8
79.9
60.2
78.8
61.6
200
167.7
108.9
165.8
111.8
163.8
114.7
161.8
117.6
159.7
120.4
157.6
123.1
300
251.6
163.4
248.7
167.8
245.7
172.1
242.7
176.3
239.6
180.5
236.4
184.7
400
335.5
217.8
331.6
223.7
327.7
229.4
323.6
235.1
319.4
240.7
315.2
246.3
500
419.3
272.3
414.5
279.6
409.6
286.8
404.5
293.9
399.3
300.9
394.0
307.8
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(123°, 237°,
(124°, 236°,
(125°, 235°,
(126°, 234°,
(127°, 233°,
(128°, 232°,
303°)
304°)
305°)
306°)
307°)
308°)
57°
5 Pt. 56°
55°
54°
4fPt. 53°
52°
The 3-Pt. or 34° Courses are : N.E. by N., N.W. by N., S.E. by S., S.W. by S,
Table 1. Traverse Table
165
33°
3 Pt. 34°
35°
36°
3iPt. 37°
38°
(147°, 213°,
5(146°, 214°,
(145°, 215°,
(144°, 216°,
(143°, 217°,
(142°, 218°,
DlST.
827°)
326°)
325°)
324°)
323°)
322°)
Lat.
Dep.
iLat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
42.8
27.8
42.3
28.5
41.8
29.3
41.3
30.0
40.7
30.7
40.2
31.4
52
43.6
28.3
43.1
29.1
42.6
29.8
42.1
30.6
41.5
31.3
41.0
32.0
53
44.4
28.9
43.9
29.6
43.4
30.4
42.9
31.2
42.3
31.9
41.8
32.6
54
45.3
29.4
44.8
30.2
44.2
31.0
43.7
31.7
43.1
32.5
42.6
33.2
55
46.1
30.0
45.6
30.8
45.1
31.5
44.5
32.3
43.9
•33.1
43.3
33.9
56
47.0
30.5
46.4
31.3
45.9
32.1
45.3
32.9
44.7
33.7
44.1
34.5
57
47.8
31.0
47.3
31.9
46.7
32.7
46.1
33.5
45.5
34.3
44.9
35.1
58
48.6
31.6
48.1
32.4
47.5
33.3
46.9
34.1
46.3
34.9
45.7
35.7
59
49.5
32.1
48.9
33.0
48.3
33.8
47.7
34.7
47.1
35.5
46.5
36.3
60
50.3
32.7
49.7
33.6
49.1
34.4
48.5
35.3
47.9
36.1
47.3
36.9
61
51.2
33.2
50.6
34.1
50.0
35.0
49.4
35.9
48.7
36.7
48.1
37.6
62_
52.0
33.8
51.4
34.7
50.8
35.6
50.2
36.4
49.5
37.3
48.9
38.2
63
52.8
34.3
52.2
S5.2
51.6
36.1
51.0
37.0
50.3
37.9
49.6
38.8
64
53.7
34.9
53.1
35.8
52.4
36.7
51.8
37.6
51.1
38.5
50.4
39.4
65
54.5
35.4
53.9
36.3
53.2
37.3
52.6
38.2
51.9
39.1
51.2
40.0
66
55.4
35.9
54.7
36.9
54.1
37.9
53.4
38.8
52.7
39.7
52.0
40.6
67
56.2
36.5
55.5
37.5
54.9
38.4
54.2
39.4
53.5
40.3
52.8
41.2
68
57.0
37.0
56.4
38.0
55.7
39.0
55.0
40.0
54.3
40.9
53.6
41.9
69
57.9
37.6
57.2
38.6
56.5
39.6
55.8
40.6
55.1
41.5
54.4
42.5
70
58.7
38.1
58.0
39.1
57.3
40.2
56.6
41.1
55.9
42.1
55.2
43.1
71
59.5
38.7
58.9
39.7
58.2
40.7
57.4
41.7
56.7
42.7
55.9
43.7
72
60.4
39.2
59.7
40.3
59.0
41.3
58.2
42.3
57.5
43.3
56.7
44.3
73
61.2
39.8
60.5
40.8
59.8
41.9
59.1
42.9
58.3
43.9
57.5
44.9
74
62.1
40.3
61.3
41.4
60.6
42.4
59.9
43.5
59.1
44.5
58.3
45.6
75
62.9
40.8
62.2
41.9
61.4
43.0
60.7
44.1
59.9
45.1
59.1
46.2
76
63.7
41.4
63.0
42.5
62.3
43.6
61.5
44.7
60.7
45.7
59.9
46.8
77
64.6
41.9
63.8
43.1
63.1
44.2
62.3
45.3
61.5
46.3
60.7
47.4
78
65.4
42.5
64.7
43.6
63.9
44.7
63.1
45.8
62.3
46.9
61.5
48.0
79
66.3
43.0
65.5
44.2
64.7
45.3
63.9
46.4
63.1
47.5
62.3
48.6
80
67.1
43.6
66.3
44.7
65.5
45.9
64.7
47.0
63.9
48.1
63.0
49.3
81
67.9
44.1
67.2
45.3
66.4
46.5
65.5
47.6
64.7
48.7
63.8
49.9
82
68.8
44.7
68.0
45.9
67.2
47.0
66.3
48.2
65.5
49.3
64.6
50.5
83
69.6
45.2
68.8
46.4
68.0
47.6
67.1
48.8
66.3
50.0
65.4
51.1
84
70.4
45.7
69.6
47.0
68.8
48.2
68.0
49.4
67.1
50.6
66.2
51.7
85
71.3
46.3
70.5
47.5
69.6
48.8
68.8
50.0
67.9
51.2
67.0
52.3
86
72.1
46.8
71.3
48.1
70.4
49.3
69.6
50.5
68.7
51.8
67.8
52.9
87
73.0
47.4
72.1
48.6
71.3
49.9
70.4
51.1
69.5
52.4
68.6
53.6
88
73.8
47.9
73.0
49.2
72.1
50.5
71.2
51.7
70.3
53.0
69.3
54.2
89
74.6
48.5
73.8
49.8
72.9
51.0
72.0
52.3
71.1
53.6
70.1
54.8
90
75.5
49.0
74.6
50.3
73.7
51.6
72.8
52.9
71.9
54.2
70.9
55.4
91
76.3
49.6
75.4
50.9
74.5
52.2
73.6
53.5
72.7
54.8
71.7
56.0
92
77.2
50.1
76.3
51.4
75.4
52.8
74.4
54.1
73.5
55.4
72.5
56.6
93
78.0
50.7
77.1
52.0
76.2
53.3
75.2
54.7
74.3
56.0
73.3
57.3
94
78.8
51.2
77.9
52.6
77.0
53.9
76.0
55.3
75.1
56.6
74.1
57.9
95
79.7
51.7
78.8
53.1
77.8
54.5
76.9
55.8
75.9
57.2
74.9
58.5
96
80.5
52.3
79.6
53.7
78.6
55.1
77.7
56.4
76.7
57.8
75.6
59.1
97
81.4
52.8
80.4
54.2
79.5
55.6
78.5
57.0
77.5
58.4
76.4
59.7
98
82.2
53.4
81.2
54.8
80.3
56.2
79.3
57.6
78.3
59.0
77.2
60.3
99
83.0
53.9
82.1
55.4
81.1
56.8
80.1
58.2
79.1
59.6
78.0
61.0
100
83.9
54.5
82.9
55.9
81.9
57.4
80.9
58.8
79.9
60.2
78.8
61.6
600
503.2
326.8
497.4
335.5
491.5
344.1
485.4
352.7
479.2
361.1
472.8
369.4
700
587.0
381.3
580.3
391.4
573.5
401.5
566.2
411.4
559.0
421.3
551.6
430.8
800
671.0
435.7
663.3
447.4
655.4
458.8
647.3
470.2
638.9
481.5
630.4
492.5
900
754.8
490.1
746.1
503.2
737.2
516.2
728.1
528.9
718.6
541.7
709.1
554.0
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(123°, 237°,
(124°, 236°,
(125°, 235°,
(126°, 234°,
(127°, 233°,
(128°, 232°.
303°)
304")
305°)
306°)
307°)
308°)
57°
5 Pt. 56°
55° .
54°
4J Pt. 53°
52°
The 5-Pt. or 56° Courses are : N.E. by E., S.E. by E., N.W. by W., S.W. by W.
166
Table 1. Traverse Table
3| Pt. 39°
40
41°
3f Pt. 42°
43°
44°
4 Pt. 45°
(141°, 219°,
(140°, 220°,
(139°, 221°,
(138°, 222°,
(137°, 223°,
(136°, 224°,
(135°, 225°,
DlST.
321°)
320°)
319°)
318°)
317°)
316°)
315°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
1
0.8
0.6
0.8
0.6
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
2
1.6
1.3
1.5
1.3
1.5
1.3
1.5
1.3
1.5
1.4
1.4
1.4
1.4
1.4
3
2.3
1.9
2.3
1.9
2.3
2.0
2.2
2.0
2.2
2.0
2.2
2.1
2.1
2.1
4
3.1
2.5
3.1
2.6
3.0
2.6
3.0
2.7
2.9
2.7
2.9
2.8
2.8
2.8
5
3.9
3.1
3.8
3.2
3.8
3.3
3.7
3.3
3.7
3.4
3.6
3.5
3.5
3.5
6
4.7
3.8
4.6
3.9
4.5
3.9
4.5
4.0
4.4
4.1
4.3
4.2
4.2
4.2
7
5.4
4.4
5.4
4.5
5.3
4.6
5.2
4.7
5.1
4.8
5.0
4.9
4.9
4.9
8
6.2
5.0
6.1
5.1
6.0
5.2
5.9
5.4
5.9
5.5
5.8
5.6
5.7
5.7
9
7.0
5.7
6.9
5.8
6.8
5.9
6.7
6.0
6.6
6.1
6.5
6.3
6.4
6.4
10
7.8
6.3
7.7
6.4
7.5
6.6
7.4
6.7
7.3
6.8
7.2
6.9
7.1
7.1
11
8.5
6.9
8.4
7.1
8.3
7.2
8.2
7.4
8.0
7.5
7.9
7.6
7.8
7.8
12
9.3
7.6
9.2
7.7
9.1
7.9
8.9
8.0
8.8
8.2
8.6
8.3
8.5
8.5
13
10.1
8.2
10.0
8.4
9.8
8.5
9.7
8.7
9.5
8.9
9.4
9.0
9.2
9.2
14
10.9
8.8
10.7
9.0
10.6
9.2
10.4
9.4
10.2
9.5
10.1
9.7
9.9
9.9
15
11.7
9.4
11.5
g.e
11.3
9.8
11.1
10.0
11.0
10.2
10.8
10.4
10.6
10.6
16
12.4
10.1
12.3
10.3
12.1
10.5
11.9
10.7
11.7
10.9
11.5
11.1
11.3
11.3
17
13.2
10.7
13.0
10.9
12.8
11.2
12.6
11.4
12.4
11.6
12.2
11.8
12.0
12.0
18
14.0
11.3
13.8
11.6
13.6
11.8
13.4
12.0
13.2
12.3
12.9
12.5
12.7
12.7
19
14.8
12.0
14.6
12.2
14.3
12.5
14.1
12.7
13.9
13.0
13.7
13.2
13.4
13.4
20
15.5
12.6
15.3
12.9
15.1
13.1
14.9
13.4
14.6
13.6
14.4
13.9
14.1
14.1
21
16.3
13.2
16.1
13.5
15.8
13.8
15.6
14.1
15.4
14.3
15.1
14.6
14.8
14.8
22
17.1
13.8
16.9
14.1
16.6
14.4
16.3
14.7
16.1
15.0
15.8
15.3
15.6
15.6
23
17.9
14.5
17.6
14.8
17.4
15.1
17.1
15.4
16.8
15.7
16.5
16.0
16.3
16.3
24
18.7
15.1
18.4
15.4
18.1
15.7
17.8
16.1
17.6
16.4
17.3
16.7
17.0
17.0
25
19.4
15.7
19.2
16.1
18.9
16.4
18.6
16.7
18.3
17.0
18.0
17.4
17.7
17.7
26
20.2
16.4
19.9
16.7
19.6
17.1
19.3
17.4
190
17.7
18.7
18.1
18.4
18.4'
27
21.0
17.0
20.7
17.4
20.4
17.7
20.1
18.1
19.7
18.4
19.4
18.8
19.1
19.1
28
21.8
17.6
21.4
18.0
21.1
18.4
20.8
18.7
20.5
19.1
20.1
19.5
19.8
19.8
29
22.5
18.3
22.2
18.6
21.9
19.0
21.6
19.4
21.2
19.8
20.9
20.1
20.5
20.5
30
23.3
18.9
23.0
19.3
22.6
19.7
22.3
20.1
21.9
20.5
21.6
20.8
21.2
21.2
31
24.1
19.5
23.7
19.9
23.4
20.3
23.0
20.7
22.7
21.1
22.3
21.5
21.9
21.9
32
24.9
20.1
24.5
20.6
24.2
21.0
23.8
21.4
23.4
21.8
23.0
22.2
22.6
22.6
33
25.6
20.8
25.3
21.2
24.9
21.6
24.5
22.1
24.1
22.5
23.7
22.9
23.3
23.3
34
26.4
21.4
26.0
21.9
25.7
22.3
25.3
22.8
24.9
23.2
24.5
23.6
24.0
24.0
35
27.2
22.0
26.8
22.5
26.4
23.0
26.0
23.4
25.6
23.9
25.2
24.3
24.7
24.7
36
28.0
22.7
27.6
23.1
27.2
23.6
26.8
24.1
26.3
24.6
25.9
25.0
25.5
25.5
37
28.8
23.3
28.3
23.8
27.9
24.3
27.5
24.8
27.1
25.2
26.6
25.7
26.2
26.2
38
29.5
23.9
29.1
24.4
28.7
24.9
28.2
25.4
27.8
25.9
27.3
26.4
26.9
26.9
39
30.3
24.5
29.9
25.1
29.4
25.6
29.0
26.1
28.5
26.6
28.1
27.1
27.6
27.6
40
31.1
25.2
30.6
25.7
30.2
26.2
29.7
26.8
29.3
27.3
28.8
27.8
28.3
28.3
41
31.9
25.8
31.4
26.4
30.9
26.9
30.5
27.4
30.0
28.0
29.5
28.5
29.0
29.0
42
32.6
26.4
32.2
27.0
31.7
27.6
31.2
28.1
30.7
28.6
30.2
29.2
29.7
29.7
43
33.4
27.1
32.9
27.6
32.5
28.2
32.0
28.8
31.4
29.3
30.9
29.9
30.4
30.4
44
34.2
27.7
33.7
28.3
33.2
28.9
32.7
29.4
32.2
30.0
31.7
30.6
31.1
31.1
45
35.0
28.3
34.5
28.9
34.0
29.5
33.4
30.1
32.9
30.7
32.4
31.3
31.8
31.8
46
35.7
28.9
35.2
29.6
34.7
30.2
34.2
30.8
33.6
31.4
33.1
32.0
32.5
32.5
47
36.5
29.6
36.0
30.2
35.5
30.8
34.9
31.4
34.4
32.1
33.8
32.6
33.2
33.2
48
37.3
30.2
36.8
30.9
36.2
31.5
35.7
32.1
35.1
32.7
34.5
33.3
33.9
33.9
49
38.1
30.8
37.5
31.5
37.0
32.1
36.4
32.8
35.8
33.4
35.2
34.0
34.6
34.6
50
38.9
31.5
38.3
32.1
37.7
32.8
37.2
33.5
36.6
34.1
36.0
34.7
35.4
35.4
100
77.7
62.9
76.6
64.3
75.5
65.6
74.3
66.9
73.1
68.2
71.9
69.5
70.7
70.7
200
155.4
125.9
153.2
128.6
150.9
131.2
148.6
133.8
146.3
136.4
143.9
138.9
141.4
141.4
300
233.1
188.8
229.8
192.8
226.4
196.8
222.9
200.7
219.4
204.6
215.8
208.4
212.1
212.1
400
310.9
251.7
306.4
257.1
301.9
262.4
297.3
267.7
292.6
272.8
287.7
277.9
282.8
282.8
500
388.6
314.7
383.0
321.4
377.3
328.0
371.6
334.6
365.7
341.0
359.7
347.3
353.5
353.5
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(129°, 231°,
(130°, 230°,
(131°, 229°,
(132°, 228°,
(133°, 227°,
(134°, 226°,
(135°, 225°,
309°)
310°)
311°
312°)
313°)
314°)
315°)
4| Pt. 51°
50°
49°
41 Pt. 48°
47°
46°
4 Pt. 45°
The 4-Pt. or 45° Courses are : N.E., N.W., S.E., S.W.
Table 1. Traverse Table
167
3^ Pt, 39°
40°
41°
3f Pt. 42°
43°
44°
4 Pt. 45°
(141°, 219°,
(140°, 220°,
(139°, 221°,
(138°, 222°,
(137°, 223°,
(136°, 224°,
(135°, 225°,
DlST.
321°)
320°)
319°)
318°)
- 317°)
316°)
315°)
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
51
39.6
32.1
39.1
32.8
38.5
33.5
37.9
34.1
37.3
34.8
36.7
35.4
36.1
36.1
52
40.4
32.7
39.8
33.4
39.2
34.1
38.6
34.8
38.0
35.5
37.4
36.1
36.8
36.8
53
41.2
33.4
40.6
34.1
40.0
34.8
39.4
35.5
38.8
36.1
38.1
36.8
37.5
37.5
54
42.0
34.0
41.4
34.7
40.8
35.4
40.1
36.1
39.5
36.8
38.8
37.5
38.2
38.2
55
42.7
34.6
42.1
35.4
41.5
36.1
40.9
36.8
40.2
37.5
39.6
38.2
38.9
38.9
56
43.5
35.2
42.9
36.0
42.3
36.7
41.6
37.5
41.0
38.2
40.3
38.9
39.6
39.6
57
44.3
35.9
43.7
36.6
43.0
37.4
42.4
38.1
41.7
38.9
41.0
39.6
40.3
40.3
58
45.1
36.5
44.4
37.3
43.8
38.1
43.1
38.8
42.4
39.6
41.7
40.3
41.0
41.0
59
45.9
37.1
45.2
37.9
44.5
38.7
43.8
39.5
43.1
40.2
42.4
41.0
41.7
41.7
60
46.6
37.8
46.0
38.6
45.3
39.4
44.6
40.1
43.9
40.9
43.2
41.7
42.4
42.4
61
47.4
38.4
46.7
39.2
46.0
40.0
45.3
40.8
44.6
41.6
43.9
42.4
43.1
43.1
62
48.2
39,0
4L£
39.9
46.8
40.7
46.1
41".5
45.3
42.3
44.6
43.1
43.8
43.8
V63
49.0
39.6
18.3
40.5
47.5
41.3
46.8
42.2
46.1
43.0
45.3
43.8
44.5
44.5
64
49.7
40.3
49!0
41.1
48.3
42.0
47.6
42.8
46.8
43.6
46.0
44.5
45.3
45.3
65
50.5
40.9
49.8
41.8
49.1
42.6
48.3
43.5
47.5
44.3
46.8
45.2
46.0
46.0
66
51.3
41.5
50.6
42.4
49.8
43.3
49.0
44.2
48.3
45.0
47.5
45.8
46.7
46.7
67
52.1
42.2
51.3
43.1
50.6
44.0
49.8
44.8
49.0
45.7
48.2
46.5
47.4
47.4
68
52.8
42.8
52.1
43.7
51.3
44.6
50.5
45.5
49.7
46.4
48.9
47.2
48.1
48.1
69
53.6
43.4
52.9
44.4
52.1
45.3
51.3
46.2
50.5
47.1
49.6
47.9
48.8
48.8
70
54.4
44.1
53.6
45.0
52.8
45.9
52.0
46.8
51.2
47.7
50.4
48.6
49.5
49.5
71
55.2
44.7
54.4
45.6
53.6
46.6
52.8
47.5
51.9
48.4
51.1
49.3
50.2
50.2
72
56.0
45.3
55.2
46.3
54.3
47.2
53.5
48.2
52.7
49.1
51.8
50.0
50.9
50.9
73
56.7
45.9
55.9
46.9
55.1
47.9
54.2
48.8
53.4
49.8
52.5
50.7
51.6
51.6
74
57.5
46.6
56.7
47.6
55.8
48.5
55.0
49.5
54.1
50.5
53.2
51.4
52.3
52.3
75
58.3
47.2
57.5
48.2
56.6
49.2
55.7
50.2
54.9
51.1
54.0
52.1
53.0
53.0
76
59.1
47.8
58.2
48.9
57.4
49.9
56.5
50.9
55.6
51.8
54.7
52.8
53.7
53.7
77
59.8
48.5
59.0
49.5
58.1
50.5
57.2
51.5
56.3
52.5
55.4
53.5
54.4
54.4
78
60.6
49.1
59.8
50.1
58.9
51.2
58.0
52.2
57.0
53.2
56.1
54.2
55.2
55.2
79
61.4
49.7
60.5
50.8
59.6
51.8
58.7
52.9
57.8
53.9
56.8
54.9
55.9
55.9
80
62.2
50.3
61.3
51.4
60.4
52.5
59.5
53.5
58.5
54.6
57.5
55.6
56.6
56.6
81
62.9
51.0
62.0
52.1
61.1
53.1
60.2
54.2
59.2
55.2
58.3
56.3
57.3
57.3
82
63.7
51.6
62.8
52.7
61.9
53.8
60.9
54.9
60.0
55.9
59.0
57.0
58.0
58.0
83
64.5
52.2
63.6
53.4
62.6
54.5
61.7
55.5
60.7
56.6
59.7
57.7
58.7
58.7
84
65.3
52.9
64.3
54.0
63.4
55.1
62.4
56.2
61.4
57.3
60.4
58.4
59.4
59.4
85
66.1
53.5
65.1
54.6
64.2
55.8
63.2
56.9
62.2
58.0
61.1
59.0
60.1
60.1
86
66.8
54.1
65.9
55.3
64.9
56.4
63.9
57.5
62.9
58.7
61.9
59.7
60.8
60.8
87
67.6
54.8
66.6
55.9
65.7
57.1
64.7
58.2
63.6
59.3
62.6
60.4
61.5
61.5
88
68.4
55.4
67.4
56.6
66.4
57.7
65.4
58.9
64.4
60.0
63.3
61.1
62.2
62.2
89
69.2
56.0
68.2
57.2
67.2
58.4
66.1
59.6
65.1
60.7
64.0
61.8
62.9
62.9
90
69.9
56.6
68.9
57.9
67.9
59.0
66.9
60.2
65.8
61.4
64.7
62.5
63.6
63.6
91
70.7
57.3
69.7
58.5
68.7
59.7
67.6
60.9
66.6
62.1
65.5
63.2
64.3
64.3
92
71.5
57.9
70.5
59.1
69.4
60.4
68.4
61.6
67.3
62.7
66.2
63.9
65.1
65.1
93
72.3
58.5
71.2
59.8
70.2
61.0
69.1
62.2
68.0
63.4
66.9
64.6
65.8
65.8
94
73.1
59.2
72.0
60.4
70.9
61.7
69.9
62.9
68.7
64.1
67.6
65.3
66.5
66.5
95
73.8
59.8
72.8
61.1
71.7
62.3
70.6
63.6
69.5
64.8
68.3
66.0
67.2
67.2
96
74.6
60.4
73.5
61.7
72.5
63.0
71.3
64.2
70.2
65.5
69.1
66.7
67.9
67.9
97
75.4
61.0
74.3
62.4
73.2
63.6
72.1
64.9
70.9
66.2
69.8
67.4
68.6
68.6
98
76.2
61.7
75.1
63.0
74.0
64.3
72.8
65.6
71.7
66.8
70.5
68.1
69.3
69.3
99
76.9
62.3
75.8
63.6
74.7
64.9
73.6
66.2
72.4
67.5
71.2
68.8
70.0
70.0
100
77.7
62.9
76.6
64.3
75.5
65.6
74.3
66.9
73.1
68.2
71.9
69.5
70.7
70.7
600
466.3
377.6
459.6
385.7
452.8
393.6
445.9
401.5
438.8
409.2
431.6
416.8
424.3
424.3
700
543.9
440.6
536.3
450.0
528.3
459.2
520.2
468.4
511.9
477.4
503.5
486.3
495.0
495.0
800
621.8
503.5
613.0
514.2
603.9
524.8
594.6
535.3
585.1
545.6
575.4
555.8
565.7
565.7
900
699.3
566.3
689.5
578.5
679.2
590.3
668.8
602.2
658.2
613.8
647.3
625.2
636.3
636.3
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
(129°, 231°,
(130°, 230°,
(131°, 229°,
(132°, 228°,
(133°, 227°,
(134°, 226°,
(135°, 225°,
309°)
310°)
311°)
312°)
313°)
314°)
315°)
4| Pt. 51°
50°
49°
41 Pt. 48°
47°
46°
4 Pt. 45°
The 4-Pt. or 45° Courses are : N.E., N.W., S.E., S.W.
168
Table 2
To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIFP.
MIDDLE LATITUDE
OR
DEP.
1°
2°
3°
4°
5°
6°
7°
8°
9°
10°
11°
12°
13°
14°
15°
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
4
00
00
00
0.0
00
00
00
0.0
00
0.1
01
01
0.1
0.1
0 1
5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.1
0.2
6
00
00
0.0
00
00
00
00
0 1
0 1
0.1
01
0 1
0?
0.2
02
7
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
8
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
9
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
10
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
11
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
12
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
13
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
14
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
15
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
16
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
0.5
0.5
17
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.6
18
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.5
0.6
19
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.6
0.6
20
0.0
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.4
0.4
0.5
0.6
0.7
21
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.5
0.6
0.7
22
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.7
23
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.8
24
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.4
0.4
0.5
0.6
0.7
0.8
25
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.5
0.6
0.7
0.9
26
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.8
0.9
27
0.0
0.0
0.0
0.1
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.8
0.9
28
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.8
1.0
29
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.4
0.5
0.6
0.7
0.9
1.0
30
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
31
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.1
32
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0
1.1
33
0.0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0
1.1
34
0.0
0.0
0.0
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.6
0.7
0.9
1.0
1.2
35
0.0
0.0
0.0
0.1
0.1
0.2
0.3
0.3
0.4
0.5
0.6
0.8
0.9
1.0
1.2
36
0.0
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.4
0.5
0.7
0.8
0.9
1.1
1.2
37
0.0
0.0
0.1
0.1
0.1
0.2
0.3
0.4
0,5
0.6
0.7
0.8
0.9
1.1
1.3
38
0.0
0.0
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0
1.1
1.3
39
0.0
0.0
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.0
1.2
1.3
40
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.0
1.2
1.4
41
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.8
0.9
1.1
1.2
1.4
42
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.6
0.8
0.9
1.1
1.2
1.4
43
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.7
0.8
0.9
1.1
1.3
1.5
44
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.1
1.3
1.5
45
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.4
0.6
0.7
0.8
1.0
1.2
1.3
1.5
46
0.0
0.0
0.1
0.1
0.2
0.3
0.3
0.4
0.6
0.7
0.8
1.0
1.2
1.4
1.6
47
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.0
1.2
1.4
1.6
48
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.0
1.2
1.4
1.6
49
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.9
1.1
1.3
1.5
1.7
50
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.9
1.1
1.3
1.5
1.7
100
0.0
0.1
0.1
0.2
0.4
0.5
0.7
1.0
1.2
1.5
1.8
2.2
2.6
3.0
3.4
200
0.0
0.1
0.3
0.5
0.8
1.1
1.5
1.9
2.5
3.0
3.7
4.4
5.1
5.9
6.8
300
0.0
0.2
0.4
0.7
1.1
1.6
2.2
2.9
3.7
4.6
5.5
6.6
7.7
8.9
10.2
400
0.1
0.2
0.6
1.0
1.5
2.2
3.0
3.9
4.9
6.1
7.4
8.7
10.2
11.9
13.7
500
0.1
0.3
0.7
1.2
1.9
2.7
3.7
4.9
6.2
7.6
9.2
10.9
12.8
14.9
17.0
1.00
1.00
1.00
1.00
1.00
1.01
1.01
1.01
1.01
1.02
1.02
1.02
1.03
1.03
1.04
FACTOB •
To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY
FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP.
Table 2
169
To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIFF.
MIDDLE LATITUDE
OR
DEP.
1°
2°
3°
4°
5°
6°
7°
8°
9°
10°
11°
12°
13°
14°
15°
51
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.8
0.9
1.1
1.3
1.5
1.7
52
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1.0
1.1
1.3
1.5
1.8
53
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.2
1.4
1.6
1.8
54
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.2
1.4
1.6
1.8
55
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.2
1.4
1.6
1.9
56
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.0
1.2
1.4
1.7
1.9
57
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.0
1.2
1.5
1.7
1.9
58
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.1
1.3
1.5
1.7
2.0
59
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.1
1.3
1.5
1.8
2.0
60
0.0
0.0
0.1
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.1
1.3
1.5
1.8
2.0
61
0.0
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
0.9
1.1
1.3
1.6
1.8
2.1
62
0.0
0.0
0.1
0.2
0.2
0.3
0.5
0.6
0.8
0.9
1.1
1.4
1.6
1.8
2.1
63
0.0
0.0
0.1
0.2
0.2
0.3
0.5
0.6
0.8
1.0
1.2
1.4
1.6
1.9
2.1
64
0.0
0.0
0.1
0.2
0.2
0.4
0.5
0.6
0.8
1.0
1.2
1.4
1.6
1.9
2.2
65
0.0
0.0
0.1
0.2
0.2
0.4
0.5
0.6
0.8
1.0
1.2
1.4
1.7
1.9
2.2
66
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1.0
1.2
1.4
1.7
2.0
2.2
67
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.2
1.5
1.7
2.0
2.3
68
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.2
1.5
1.7
2.0
2.3
69
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.8
1.0
1.3
1.5
1.8
2.0
2.4
70
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.1
1.3
1.5
1.8
2.1
2.4
71
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.1
1.3
1.6
1.8
2.1
2.4
72
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.1
1.3
1.6
1.8
2.1
2.5
73
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.7
0.9
1.1
1.3
1.6
1.9
2.2
2.5
74
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.1
1.4
1.6
1.9
2.2
2.5
75
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.1
1.4
1.6
1.9
2.2
2.6
76
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.7
0.9
1.2
1.4
1.7
1.9
2.3
2.6
77
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.7
0.9'
1.2
1.4
1.7
2.0
2.3
2.6
78
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.4
1.7
2.0
2.3
2.7
79
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.5
1.7
2.0
2.3
2.7
80
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.5
1.7
2.1
2.4
2.7
81
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.5
1.8
2.1
2.4
2.8
82
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
1.0
1.2
1.5
1.8
2.1
2.4
2.8
83
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
1.0
1.3
1.5
1.8
2.1
2.5
2.8
84
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
1.0
1.3
1.5
1.8
2.2
2.5
2.9
85
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
1.0
1.3
1.6
1.9
2.2
2.5
2.9
86
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
1.1
1.3
1.6
1.9
2.2
2.6
2.9
87
0.0
0.1
0.1
0.2
0.3
0.5
0.6
0.8
1.1
1.3
1.6
1.9
2.2
2.6
3.0
88
0.0
0.1
0.1
0.2
0.3
0.5
0.7
0.9
1.1
1.3
1.6
1.9
2.3
2.6
3.0
89
0.0
0.1
0.1
0.2
0.3
0.5
0.7
0.9
1.1
1.4
1.6
1.9
2.3
2.6
3.0
90
0.0
0.1
0.1
0.2
0.3
0.5
0.7
0.9
1.1
1.4
1.7
2.0
2.3
2.7
3.1
91
0.0
0.1
0.1
0.2
0.3
0.5
0.7
0.9
1.1
1.4
1.7
2.0
2.3
2.7
3.1
92
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.1
1.4
1.7
2.0
2.4
2.7
3.1
93
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.1
1.4
1.7
2.0
2.4
2.8
3.2
94
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.2
1.4
1.7
2.1
2.4
2.8
3.2
95
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.2
1.4
1.7
2.1
2.4
2.8
3.2
96
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.2
1.5
1.8
2.1
2.5
2.9
3.3
97
0.0
0.1
0.1
0.2
0.4
0.5
0.7
0.9
1.2
1.5
1.8
2.1
2.5
2.9
3.3
98
0.0
0.1
0.1
0.2
0.4
0.5
0.7
1.0
1.2
1.5
1.8
2.1
2.5
2.9
3.3
99
0.0
0.1
0.1
0.2
0.4
0.5
0.7
1.0,
1.2
1.5
1.8
2.2
2.5
2.9
3.4
100
0.0
0.1
0.1
0.2
0.4
0.5
0.7
1.0
1.2
1.5
1.8
2.2
2.6
3.0
3.4
600
0.1
0.4
0.8
1.4
2.3
3.3
4.5
5.8
7.4
9.1
10.0
13.1
15.4
17.8
20.5
700
0.2
0.5
1.0
1.8
2.8
3.9
5.1
6.7
8.7
10.5
12.9
15.3
17.9
20.8
23.9
800
0.2
0.5
1.1
2.0
3.1
4.4
5.9
7.7
9.8
12.1
14.8
17.5
20.6
23.8
27.3
900
0.3
0.7
1.4
2.4
3.6
5.0
6.7
8.7
11.2
13.7
16.7
19.8
23.2
26.8
30.8
1.00
1.00
1.00
1.00
1.00
1.01
1.01
1.01
1.01
1.02
1.02
1.02
1.03
1.03
1.04
FACTOR
To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP.
170
Table 2
To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIFF.
MIDDLE LATITUDE
DEP.
16°
17°
18°
19°
20°
21°
22°
23°
24°
25°
26°
27°
28°
1
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
3
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.4
4
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.5
5
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
6
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
7
0.3
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
8
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.9
0.9
9
0.3
0.4
0.4
0.5
0.5
0.6
0.7
0.7
0.8
0.8
0.9
1.0
1.1
10
0.4
0.4
0.5
0.5
0.6
0.7
0.7
0.8
0.9
0.9
1.0
1.1
1.2
11
0.4
0.5
0.5 '
0.6
0.7
0.7
0.8
0.9
1.0
1.0
1.1
1.2
1.3
12
05
0.5
0.6
0.7
0.7
0.8
0.9
1.0
1.0
1.1
1.2
1.3
1 4
13
0.5
0.6
0.6
0.7
0.8
0.9
0.9
1.0
1.1
1.2
1.3
1.4
1.5
14
0.5
0.6
0.7
0.8
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
15
0.6
0.7
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.8
16
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.9
17
0.7
0.7
0.8
0.9
1.0
1.1
1.2
1.4
1.5
1.6
1.7
1.9
2.0
18
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.6
1.7
1.8
2.0
2.1
19
0.7
0.8
0.9
1.0
1.1
1.3
1.4
1.5
1.6
1.8
1.9
2.1
2.2
20
0.8
0.9
1.0
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.0
2.2
2.3
21
0.8
0.9
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.1
2.3
2.5
22
0.9
1.0
1.1
1.2
1.3
1.5
1.6
1.7
1.9
2.1
2.2
2.4
2.6
23
0.9
1.0
1.1
1.3
1.4
1.5
1.7
1.8
2.0
2.2
2.3
2.5
2.7
24
0.9
1.0
1.2
1.3
1.4
1.6
1.7
1.9
2.1
2.2
2.4
2.6
2.8
25
1.0
1.1
1.2
1.4
1.5
1.7
1.8
2.0
2.2
2.3
2.5
2.7
2.9
26
1.0
1.1
1.3
1.4
1.6
1.7
1.9
2.1
2.2
2.4
2.6
2.8
3.0
27
1.0
1.2
1.3
1.5
1.6
1.8
2.0
2.1
2.3
2.5
2.7
2.9
3.2
28
1.1
1.2
1.4
1.5
1.7
1.9
2.0
2.2
2.4
2.6
2.8
3.1
3.3
29
1.1
1.3
1.4
1.6
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.2
3.4
30
1.2
1.3
1.5
1.6
1.8
2.0
2.2
2.4
2.6
2-.8
3.0
3.3
3.5
31
1.2
1.4
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.4
3.6
32
1.2
1.4
1.6
1.7
1.9
2.1
2.3
2.5
2.8
3.0
3.2
3.5
3.7
33
1.3
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.9
3.1
3.3
3.6
3.9
34
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.2
3.4
3.7
4.0
35
1.4
1.5
1.7
1.9
2.1
2.3
2.5
2.8
3.0
3.3
3.5
3.8
4.1
36
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.9
3.1
3.4
3.6
3.9
4.2
37
1.4
1.6
1.8
2.0
2.2
2.5
2.7
2.9
3.2
3.5
3.7
4.0
4.3
38
1.5
1.7
1.9
2.1
2.3
2.5
2.8
3.0
3.3
3.6
3.8
4.1
4.4
39
1.5
1.7
1.9
2.1
2.4
2.6
2.8
3.1
3.4
3.7
3.9
4.3
4.6
40
1.5
1.7
2.0
2.2
2.4
2.7
2.9
3.2
3.5
3.7
4.0
4.4
4.7
41
1.6
1.8
2.0
2.2
2.5
2.7
3.0
3.3
3.5
3.8
4.1
4.5
4.8
42
1.6
1.8
2.1
2.3
2.5
2.8
3.1
3.3
3.6
3.9
4.3
4.6
4.9
43
1.7
1.9
2.1
2.3
2.6
2.9
3.1
3.4
3.7
4.0
1.4
4.7
5.0
44
1.7
1.9
2.2
2.4
2.7
2.9
3.2
3.5
3.8
4.1
4.5
4.8
5.2
45
1.7
2.0
2.2
2.5
2.7
3.0
3.3
3.6
3.9
4.2
4.6
4.9
5.3
46
1.8
2.0
2.3
2.5
2.8
3.1
3.3
3.7
4.0
4.3
4.7
5.0
5.4
47
1.8
2.1
2.3
2.6
2.8
3.1
3.4
3.7
4.1
4.4
4.8
5.1
5.5
48
1.9
2.1
2.3
2.6
2.9
3.2
3.5
3.8
4.1
4.5
4.9
5.2
5.6
49
1.9
2.1
2.4
2.7
3.0
3.3
3.6
3.9
4.2
4.6
5.0
5.3
5.7
50
1.9
2.2
2.4
2.7
3.0
3.3'
3.6
4.0
4.3
4.7
5.1
5.4
5.9
100
3.9
4.4
4.9
5.4
6.0
6.6
7.3
7.9
8.6
9.4
10.1
10.9
11.7
200
7.7
8.7
9.8
10.9
12.1
13.3
14.6
15.9
17.3
18.7
20.2
21.8
23.4
300
11.6
13.1
14.7
16.3
18.1
19.9
21.8
23.8
25.9
28.1
30.4
32.7
35.1
400
15.5
17.5
19.6
21.8
24.1
26.6
29.1
31.8
34.6
37.5
40.5
43.6
46.9
500
19.4
21.9
24.5
27.2
30.1
33.2
36.4
39.8
43.2
46.9
50.6
54.5
58.5
1.04
1.05
1.05
1.06
1.06
1.07
1.08
1.09
1.09
1.10
1.11
1.12
1.13
FACTOR
To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY
T^ATTOR AT TTnnT nw (^nT.rnwivr Aiun Ann PRnnrrrT TO DF.P.
Table 2
171
To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIPP.
MIDDLE LATITUDE
OR
DEP.
16°
17°
18°
19°
20°
21°
22°
23°
24°
25°
26°
27°
28°
51
2.0
2.2
2.5
2.8
3.1
3.4
3.7
4.1
4.4
4.8
5.2
5.6
6.0
52
2.0
2.3
2.5
2.8
3.1
3.5
3.8
4.1
4.5
4.9
5.3
5.7
6.1
53
2.1
2.3
2.6
2.9
3.2
3.5
3.9
4.2
4.6
5.0
5.4
5.8
6.2
54
2.1
2.4
2.6
2.9
3.3
3.6
3.9
4.3
4.7
5.1
5.5
5.9
6.3
55
2.1
2.4
2.7
3.0
3.3
3.7
4.0
4.4
4.8
5.2
'5.6
6.0
6.4
56
2.2
2.4
2.7
3.1
3.4
3.7
4.1
4.5
4.8
5.2
5.7
6.1
6.6
57
2.2
2.5
2.8
3.1
3.4
3.8
4.2
4.5
4.9
5.3
5.8
6.2
6.7
58
2.2
2.5
2.8
3.2
3.5
3.9
4.2
4.6
5.0
5.4
5.9
6.3
6.8
59
2.3
2.6
2.9
3.2
3.6
3.9
4.3
4.7
5.1
5.5
6.0
6.4
6.9
60
2.3
2.6
2.9
3.3
3.6
4.0
4.4
4.8
5.2
5.6
6.1
6.5
7.0
61
2.4
2.7
3.0
3.3
3.7
4.1
4.4
4.8
5.3
5.7
6.2
6.6
7.1
62
2.4
2.7
3.0
3.4
3.7
4.1
4.5
4.9
5.4
5.8
6.3
6.8
7.3
63
2.4
2.8
3.1
3.4
3.8
4.2
4.6
5.0
5.4
5.9
6.4
6.9
7.4
64
2.5
2.8
3.1
3.5
3.9
4.3
4.7
5.1
5.5
6.0
6.5
7.0
7.5
65
2.5
2.8
3.2
3.5
3.9
4.3
4.7
5.2
5.6
6.1
6.6
7.1
7.6
66
2.6
2.9
3.2
3.6
4.0
4.4
4.8
5.2
5.7
6.2
6.7
7.2
7.7
67
2.6
2.9
3.3
3.7
4.0
4.5
4.9
5.3
5.8
6.3
6.8
7.3
7.8
68
2.6
3.0
3.3
3.7
4.1
4.5
5.0
5.4
5.9
6.4
6.9
7.4
8.0
69
2.7
3.0
3.4
3.8
4.2
4.6
5.0
5.5
6.0
6.5
7.0
7.5
8.1
70
2.7
3.1
3.4
3.8
4.2
4.6
5.1
5.6
6.1
6.6
7.1
7.6
8.2
71
2.8
3.1
3.5
3.9
4.3
4.7
5.2
5.6
6.1
6.7
7.2
7.7
8.3
72
2.8
3.1
3.5
3.9
4.3
4.8
5.2
5.7
6.2
6.7
7.3
7.8
8.4
73
2.8
3.2
3.6
4.0
4.4
4.8
5.3
5.8
6.3
6.8
7.4
8.0
8.5
74
2.9
3.2
3.6
4.0
4.5
4.9
5.4
5.9
6.4
6.9
7.5
8.1
8.7
75
2.9
3.3
3.7
4.1
4.5
5.0
5.5
6.0
6.5
7.0
7.6
8.2
8.8
76
2.9
3.3
3.7
4.1
4.6
5.0
5.5
6.0
6.6
7.1
7.7
8.3
8.9
77
3.0
3.4
3.8
4.2
4.6
5.1
5.6
6.1
6.7
7.2
7.8
8.4
9.0
78
3.0
3.4
3.8
4.2
4.7
5.2
5.7
6.2
6.7
7.3
7.9
8.5
9.1
79
3.1
3.5
3.9
4.3
4.8
5.2
5.8
6.3
6.8
7.4
8.0
8.6
9.2
80
3.1
3.5
3.9
4.4
4.8
5.3
5.8
6.4
6.9
7.5
8.1
8.7
9.4
81
3.1
3.5
4.0
4.4
4.9
5.4
5.9
6.4
7.0
7.6
8.2
8.8
9.5
82
3.2
3.6
4.0
4.5
4.9
5.4
6.0
6.5
7.1
7.7
8.3
8.9
9.6
83
3.2
3.6
4.1
4.5
5.0
5.5
6.0
6.6
7.2
7.8
8.4
9.0
9.7
84
3.3
3.7
4.1
4.6
5.1
5.6
6.1
6.7
7.3
7.9
8.5
9.2
9.8
85
3.3
3.7
4.2
4.6
5.1
5.6
6.2
6.8
7.3
8.0
8.6
9.3
9.9
86
3.3
3.8
4.2
4.7
5.2
5.7
6.3
6.8
7.4
8.1
8.7
9.4
10.1
87
3.4
3.8
4.3
4.7
5.2
5.8
6.3
6.9
7.5
8.2
8.8
9.5
10.2
• 88
3.4
3.8
4.3
4.8
5.3
5.8
6.4
7.0
7.6
8.2
8.9
9.6
10.3
89
3.4
3.9
4.4
4.8
5.4
5.9
6.5
7.1
7.7
8.3
9.0
9.7
10.4
90
3.5
3.9
4.4
4.9
5.4
6.0
6.6
7.2
7.8
8.4
9.1
9.8
10.5
91
3.5
4.0
4.5
5.0
5.5
6.0
6.6
7.2
7.9
8.5
9.2
9.9
10.7
92
3.6
4.0
4.5
5.0
5.5
6.1
6.7
7.3
8.0
8.6
9.3
10.0
10.8
93
3.6
4.1
4.6
5.1
5.6
6.2
6.8
7.4
8.0
8.7
9.4
10.1
10.9
94
3.6
4.1
4.6
5.1
5.7
6.2
6.8
7.5
8.1
8.8
9.5
10.2
11.0
95
3.7
4.2
4.6
5.2
5.7
6.3
6.9
7.6
8.2
8.9
9.6
10.4
11.1
96
3.7
4.2
4.7
5.2
5.8
6.4
7.0
7.6
8.3
9.0
9.7
10.5
11.2
97
3.8
4.2
4.7
5.3
5.8
6.4
7.1
7.7
8.4
9.1
9.8
10.6
11.4
98
3.8
4.3
4.8
5.3
5.9
6.5
7.1
7.8
8.5
9.2
9.9
10.7
11.5
99
3.8
4.3
4.8
5.4
6.0
6.6
7.2
7.9
8.6
9.3
10.0
10.8
11.6
100
3.9
4.4
4.9
5.4
6.0
6.6
7.3
7.9
8.6
9.4
10.1
10.9
11.7
600
23.2
26.2
29.4
32.7
36.2
39.9
43.7
47.7
51.9
56.2
60.7
65.4
70.2
700
27.2
30.6
34.2
38.1
42.1
46.4
50.9
55.7
60.5
65.5
70.8
76.3
82.0
800
31.0
35.0
39.2
43.5
48.2
53.1
58.2
63.6
69.2
74.9
80.9
87.1
93.7
900
35.0
39.4
44.1
49.1
54.3
59.7
65.5
71.7
77.9
84.4
91.1
98.1
105.5
1.04
1.05
1.05
1.06
1.06
1.07
1.08
1.09
1.10
1.10
1.11
1.12
1.13
FACTOB
To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY
FACTOR AT FOOT OF COLUMN, AND ADD PRODUCT TO DEP.
172
Table 2
To CHANGE LONG. DIFP. INTO DEP., SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONO.
DIFF.
• MIDDLE LATITUDE
OR
DEP.
29°
30°
31°
32°
33°
34°
35°
36°
37°
38°
39°
40°
1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
2
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.5
3
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
4
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.8
0.8
0.8
0.9
0.9
5
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1.0
1.0
1.1
1.1
1.2
6
0.8
0.8
0.9
0.9
1.0
1.0
1.1
1.1
1.2
1.3
1.3
1.4
7
0.9
0.9
1.0
1.1
1.1
1.2
1.3
1.3
1.4
1.5
1.6
1.6
8
1.0
1.1
1.1
1.2
1.3
1.4
1.4
1.5
1.6
1.7
1.8
1.9
9
1.1
1.2
1.3
1.4
1.5
1.5
1.6
1.7
1.8
1.9
2.0
2.1
10
1.3
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
11
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.5
2.6
12
1.5
1.6
1.7
1.8
1.9
2.1
2.2
2.3
. 2.4
2.5
2.7
2.8
13
1.6
1.7
1.9
2.0
2.1
2.2
2.4
2.5
2.6
2.8
2.9
3.0
14
1.8
1.9
2.0
2.1
2.3
2.4
2.5
2.7
2.8
3.0
3.1
3.3
15
1.9
2.0
2.1
2.3
2.4
2.6
2.7
2.9
3.0
3.2
3.3
3.5
16
2.0
2.1
2.3
2.4
2.6
2.7
2.9
3.1
3.2.
3.4
3.6
3.7
17
2.1
2.3
2.4
2.6
2.7
2.9
3.1
3.2
3.4
3.6
3.8
4.0
18
2.3
2.4
2.6
2.7
2.9
3.1
3.3
3.4
3.6
3.8
4.0
4.2
19
2.4
2.5
2.7
2.9
3.1
3.2
3.4
3.6
3.8
4.0
4.2
4.4
20
2.5
2.7
2.9
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.5
4.7
21
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.5
4.7
4.9
22
2.8
2.9
3.1
3.3
3.5
3.8
4.0
4.2
4.4
4.7
4.9
5.1
23
2.9
3.1
3.3
3.5
3.7
3.9
4.2
4.4
4.6
4.9
5.1
5.4
24
3.0
3.2
3.4
3.6
3.9
4:1
4.3
4.6
4.8
5.1
5.3
5.6
25
3.1
3.3
3.6
3.8
4.0
4.3
4.5
4.8
5.0
5.3
5.6
5.8
26
3.3
3.5
3.7
4.0
4.2
4.4
4.7
5.0
5.2
5.5
5.8
6.1
27
3.4
3.6
3.9
4.1
4.4
4.6
4.9
5.2
5.4
5.7
6.0
6.3
28
3.5
3.8
4.0
4.3
4.5
4.8
5.1
5.3
5.6
5.9
6.2
6.6
29
3.6
3.9
4.1
4.4
4.7
5.0
5.2
5.5
5.8
6.1
6.5
6.8
30
3.8
4.0
4.3
4.6
4.8
5.1
5.4
5.7
6.0
6.4
6.7
7.0
31
3.9
4.2
4.4
4.7
5.0
5.3
5.6
5.9
6.2
6.6
6.9
7.3
32
4.0
4.3
4.6
4.9
5.2
5.5
5.8
6.1
6.4
6.8
7.1
7.5
33
4.1
4.4
4.7
5.0
5.3
5.6
6.0
6.3
6.6
7.0
7.4
7.7
34
4.3
4.6
4.9
5.2
5.5
5.8
6.1
6.5
6.8
7.2
7.6
8.0
35
4.4
4.7
5.0
5.3
5.6
6.0
6.3
6.7
7.0
7.4
7.8
8.2
36
4.5
4.8
5.1
5.5
5.8
6.2
6.5
6.9
7.2
7.6
8.0
8.4
37
4.6
5.0
5.3
5.6
6.0
6.3
6.7
7.1
7.5
7.8
8.2
8.7
38
4.8
5.1
5.4
5.8
6.1
6.5
6.9
7.3
7.7
8.1
8.5
8.9
39
4.9
5.2
5.6
5.9
6.3
6.7
7.1
7.4
7.9
8.3
8.7
9.1
40
5.0
5.4
5.7
6.1
6.5
6.8
7.2
7.6
8.1
8.5
8.9
9.4
41
5.1
5.5
5.9
6.2
6.6
7.0
7.4
7.8
8.3
8.7
9.1
9.6
42
5.3
5.6
6.0
6.4
6.8
7.2
7.6
8.0
8.5
8.9
9.4
9.8
43
5.4
5.8
6.1
6.5
6.9
7.4
7.8
8.2
8.7
9.1
9.6
10.1
44
5.5
5.9
6.3
6.7
7.1
7.5
8.0
8.4
8.9
9.3
9.8
10.3
45
5.6
6.0
6.4
6.8
7.3
7.7
8.1
8.6
9.1
9.5
10.0
10.5
46
5.8
6.2
6.6
7.0
7.4
7.9
8.3
8.8
9.3
9.8
10.3
10.8
47
5.9
6.3
6.7
7.1
7.6
8.0
8.5
9.0
9.5
10.0
10.5
11.0
48
6.0
6.4
6.9
7.3
7.7
8.2
8.7
9.2
9.7
10.2
10.7
11.2
49
6.1
6.6
7.0
7.4
7.9
8.4
8.9
9.4
9.9
10.4
10.9
11.5
50
6.3.
6.7
7.1
7.6
8.1
8.5
9.0
9.5
10.1
10.6
11.1
11.7
100
12.5
13.4
14.3
15.2
16.1
17.1
18.1
19.1
20.1
21.2
22.3
23.4
200
25.1
26.8
28.6
30.4
32.3
34.2
36.2
38.2
40.3
42.4
44.6
46.8
300
37.6
40.2
42.9
45.6
48.4
51.3
54.3
57.3
60.4
63.6
66.9
70.2
400
50.2
53.6
57.1
60.8
64.5
68.4
72.3
76.4
80.6
84.8
89.1
93.6
500
62.7
67.0
71.4
76.0
80.7
85.5
90.4
95.5
100.7
106.0
111.4
117.0
1.14
1.15
1.17
1.18
1.19
1.21
1.22
1.24
1.25
1.27
1.29
1.31
FACTOR
To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY
" ~
Table 2
173
To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
MIDDLE LATITUDE
DIFF.
OR
DEP.
29°
30°
31°
32°
33°
34°
35°
36°
37°
38°
39°
40°
51
6.4
6.8
7.3
7.7
8.2
8.7
9.2
9.7
10.3
10.8
ir.4
11.9
52
6.5
7.0
7.4
7.9
8.4
8.9
9.4
9.9
10.5
11.0
11.6
12.2
53
6.6
7.1
7.6
8.1
8.6
9.1
9.6
10.1
10.7
11.2
11.8
12.4
54
6.8
7.2
7.7
8.2
8.7
9.2
9.8
10.3
10.9
11.4
12.0
12.6
55
6.9
7.4
7.9
8.4
8.9
9.4
9.9
10.5
11.1
11.7
12.3
12.9
56
7.0
7.5
8.0
8.5
9.0
9.6
10.1
10.7
11.3
11.9
12.5
13.1
57
7.1
7.6
8.1
8.7
9.2
9.7
10.3
10.9
11.5
12.1
12.7
13.3
58
7.3
7.8
8.3
8.8
9.4
9.9
10.5
11.1
11.7
12.3
12.9
13.6
59
7.4
7.9
8.4
9.0
9.5
10.1
10.7
11.3
11.9
12.5
13.1
13.8
60
7.5
8.0
8.6
9.1
9.7
10.3
10.9
11.5
12.1
12.7
13.4
14.0
61
7.6
8.2
8.7'
9.3
9.8
10.4
11.0
11.6
12.3
12.9
13.6
14.3
62
7.8
8.3
8.9
9.4
10.0
10.6
11.2
11.8
12.5
13.1
13.8
14.5
63
7.9
8.4
9.0
9.6
10.2
10.8
11.4
12.0
12.7
13.4
14.0
14.7
64
8.0
8.6
9.1
9.7
10.3
10.9
11.6
12.2
12.9
13.6
14.3
15.0
65
8.1
8.7
9.3
9.9
10.5
11.1
11.8
12.4
13.1
13.8
14.5
15.2
66
8.3
8.8
9.4
10.0
10.6
11.3
11.9
12.6
13.3
14.0
14.7
15.4
67
8.4
9.0
9.6
10.2
10.8
11.5
12.1
12.8
13.5
14.2
14.9
15.7
68
8.5
9.1
9.7
10.3
11.0
11.6
12.3
13.0
13.7
14.4
15.2
15.9
69
8.7
9.2
9.9
10.5
11.1
11.8
12.5
13.2
13.9
14.6
15".4
16.1
70
8.8
9.4
10.0
10.6
11.3
12.0
12.7
13.4
14.1
14.8
15.6
16.4
71
8.9
9.5
10.1
10.8
11.5
12.1
12.8
13.6
14.3
15.1
15.8
16.6
72
9.0
9.6
10.3
10.9
11.6
12.3
13.0
13.8
14.5
15.3
16.0
16.8
73
9.2
9.8
10.4
11.1
11.8
12.5
13.2
13.9
14.7
15.5
16.3
17.1
74
9.3
9.9
10.6
11.2
11.9
12.7
13.4
14.1
14.9
15.7
16.5
17.3
75
9.4
10.0
10.7
11.4
12.1
12.8
13.6
14.3
15.1
15.9
16.7
17.5
76
9.5
10.2
10.9
11.5
12.3
13.0
13.7
14.5
15.3
16.1
16.9
17.8
77
9.7
10.3
11.0
11.7
12.4
13.2
13.9
14.7
15.5
16.3
17.2
18.0
78
9.8
10.5
11.1
11.9
12.6
13.3
14.1
14.9
15.7
16.5
17.4
18.2
79
9.9
10.6
11.3
12.0
12.7
13.5
14.3
15.1
15.9
16.7
17.6
18.5
80
10.0
10.7
11.4
12.2
12.9
13.7
14.5
15.3
16.1
17.0
17.8
18.7
81
10.2
10.9
11.6
12.3
13.1
13.8
14.6
15.5
16.3
17.2
18.1
19.0
82
10.3
11.0
11.7
12.5
13.2
14.0
14.8
15.7
16.5
17.4
18.3
19.2
83
10.4
11.1
11.9
12.6
13.4
14.2
15.0
15.9
16.7
17.6
18.5
19.4
84
10.5
11.3
12.0
12.8
13.6
14.4
15.2
16.0
16.9
17.8
18.7
19.7
85
10.7
11.4
12.1
12.9
13.7
14.5
15.4
16.2
17.1
18.0
18.9
19.9
86
10.8
11.5
12.3
13.1
13.9
14.7
15.6
16.4
17.3
18.2
19.2
20.1
87
10.9
11.7
12.4
13.2
14.0
14.9
15.7
16.6
17.5
18.4
19.4
20.4
88
11.0
11.8
12.6
13.4
14.2
15.0
15.9
16.8
17.7
18.7
19.6
20.6
89
11.2
11.9
12.7
13.5
14.4
15.2
16.1
17.0
17.9
18.9
19.8
20.8
90
11.3
12.1
12.9
13.7
14.5
15.4
16.3
17.2
18.1
19.1
20.1
21.1
91
11.4
12.2
13.0
13.8
14.7
15.6
16.5
17.4
18.3
19.3
20.3
21.3
92
11.5
12.3
13.1
14.0
14.8
15.7
16.6
17.6
18.5
19.5
20.5
21.5
93
11.7
12.5
13.3
14.1
15.0
15.9
16.8
17.8
18.7
19.7
20.7
21.8
94
11.8
12.6
13.4
14.3
15.2
16.1
17.0
18.0
18.9
19.9
20.9
22.0
95
11.9
12.7
13.6
14.4
15.3
16.2
17.2
18.1
19.1
20.1
21.2
22.2
96
12.0
12.9
13.7
14.6
15:5
16.4
17.4
18.3
19.3
20.4
21.4
22.5
97
12.2
13.0
13.9
14.7
15.6
16.6
17.5
18.5
19.5
20.6
21.6
22.7
98
12.3
13.1
14.0
14.9
15.8
16.8
17.7
18.7
19.7
20.8
21.8
22.9
99
12.4
13.3
14.1
15.0
16.0
16.9
17.9
18.9
19.9
21.0
22.1
23.2
100
12.5
13.4
14.3
15.2
16.1
17.1
18.1
19.1
20.1
21.2
22.3
23.4
600
75.2
80.4
85.7
91.2
96.8
102.6
108.5
114.6
120.8
127.2
133.7
140.4
700
87.8
93.9
99.9
106.4
113.0
119.7
126.5
133.8
141.0
148.4
156.1
163.7
800
100.3
107.2
114.2
121.6
129.0
136.7
144.6
152.7
161.1
169.6
178.2
187.0
900
113.0
120.7
128.6
136.8
145.2
153.9
162.8
171.9
181.4
190.9
200.7
210.5
1.14
1.15
1.17
1.18
1.19
1.21
1.22
1.24
1.25
1.27
1.29
1.31
FACTOR
To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY
T^APTnR AT F'nn'p ni? dni.TTiww A\rr> Ann PRnr»TTr"r> TT» T^TT.TV
174
Table 2
To CHANGE LONG. DIFP. INTO DEP., SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIPF.
MIDDLE LATITUDE
OR
DEP.
41°
42°
43°
44°
45;
46°
47°
48°
49°
50°
51°
1
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
2
0.5
0.5
0.5
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
3
0.7
0.8
0.8
0.8
0.9
0.9
1.0
1.0
1.0
1.1
1.1
4
1.0
1.0
1.1
1.1
1.2
1.2
1.3
1.3
1.4
1.4
15
5
1.2
1.3
1.3
1.4
1.5
1.5
1.6
1.7
1.7
1.8
1.9
6
1.5
1.5
1.6
1.7
1.8
1.8
1.9
2.0
2.1
2.1
2.2
7
1.7
1.8
1.9
2.0
2.1
2.1
2.2
2.3
2.4
2.5
2.6
8
2.0
2.1
2.1
2.2
2.3
2.4
2.5
2.6
2.8
2.9
3.0
9
2.2
2.3
2.4
2.5
2.6
2.7
2.9
3.0
3.1
3.2
3.3
10
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.6
3.7
11
2.7
2.8
3.0
3.1
3.2
3.4
3.5
3.6
3.8
3.9
4.1
12
2.9
3.1
3.2
3.4
3.5
3.7
3.8
4.0
4.1
4.3
4.4
13
3.2
3.3
3.5
3.6
3.8
4.0
4.1
4.3
4.5
4.6
4.8
14
3.4
3.6
3.8
3.9
4.1
4.3
4.5
4.6
4.8
5.0
5.2
15
3.7
3.9
4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
16
3.9
4.1
4.3
4.5
4.7
4.9
5.1
5.3
5.5
5.7
5.9
17
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.1
6.3
18
4.4
4.6
4.8
5.1
5.3
5.5
5.7
6.0
6.2
6.4
6.7
19
4.7
4.9
5.1
5.3
5.6
5.8
6.0
6.3
6.5
6.8
7.0
20
4.9
5.1
5.4
5.6
5.9
6.1
6.4
6.6
6.9
7.1
7.4
21
5.2
5.4
5.6
5.9
6.2
6.4
6.7
6.9
7.2
7.5
7.8
22
5.4
5.7
5.9
6.2
6.4
6.7
7.0
7.3
7.6
7.9
8.2
23
5.6
5.9
6.2
6.5
6.7
7.0
7.3
7.6
7.9
8.2
8.5
24
5.9
6.2
6.4
6.7
7.0
7.3
7.6
7.9
8.3
8.6
8.9
25
6.1
6.4
6.7
7.0
7.3
7.6
8.0
8.3
8.6
8.9
9.3
26
64
6.7
7.0
7.3
7.6
7.9
8.3
8.6
8.9
9.3
96
27
6.6
6.9
7.3
7.6
7.9
8.2
8.6
8.9
9.3
9.6
10.0
28
6.9
7.2
7.5
7.9
8.2
8.5
8.9
9.3
9.6
10.0
10.4
29
7 1
7.4
7.8
8.1
8.5
8.9
9.2
9.6
10.0
10.4
107
30
7.4
7.7
8.1
8.4
8.8
9.2
9.5
9.9
10.3
10.7
11.1
31
7.6
8.0
8.3
8.7
9.1
9.5
9.9
10.3
10.7
11.1
11.5
32
7.8
8.2
8.6
9.0
9.4
9.8
10.2
10.6
11.0
11.4
11.9
33
8.1
8.5
8.9
9.3
9.7
10.1
10.5
10.9
11.4
11.8
12.2
34
8.3
8.7
9.1
9.5
10.0
10.4
10.8
11.2
11.7
12.1
12.6
35
8.6
9.0
9.4
9.8
10.3
10.7
11.1
11.6
12.0
12.5
13.0
36
8.8
9.2
9.7
10.1
10.5
11.0
11.4
11.9
12.4
12.9
13.3
37
9.1
9.5
9.9
10.4
10.8
11.3
11.8
12.2
12.7
13.2
13.7
38
9.3
9.8
10.2
10.7
11.1
11.6
12.1
12.6
13.1
13.6
14.1
39
9.6
10.0
10.5
10.9
11.4
11.9
12.4
12.9
13.4
13.9
14.5
40
9.8
10.3
10..7
11.2
11.7
12.2
12.7
13.2
13.8
14.3
14.8
41
10.1
10.5
11.0
11.5
12.0
12.5
13.0
13.6
14.1
14.6
15.2
42
10.3
10.8
11.3
11.8
12.3
12.8
13.4
13.9
14.4
15.0
15.6
43
10.5
11.0
11.6
12.1
12.6
13.1
13.7
14.2
14.8
15.4
15.9
44
10.8
11.3
11.8
12.3
12.9
13.4
14.0
14.6
15.1
15.7
16.3
45
11.0
11.6
12.1
12.6
13.2
13.7
14.3
14.9
15.5
16.1
16.7
46
11.3
11.8
12.4
12.9
13.5
14.0
14.6
15.2
15.8
16.4
17.1
47
11.5
12.1
12.6
13.2
13.8
14.4
14.9
15.6
16.2
16.8
17.4
48
11.8
12.3
12.9
13.5
14.1
14.7
15.3
15.9
16.5
17.1
17.8
49
12.0
12.6
13.2
13.8
14.4
15.0
15.6
16.2
16.9
17.5
18.2
50
12.3
12.8
13.4
14.0
14.6
15.3
15.9
16.5
17.2
17.9
18.5
100
24.5
25.7
26.9
28.1
29.3
30.5
31.8
33.1
34,4
35.7
37.1
200
49.1
51.4
53.7
56.1
58.6
61.1
63.6
66.2
68.8
71.4
74.1
300
73.6
77.1
80.6
84.2
87.9
91.6
95.4
99.3
103.2
107.2
111.2
400
98.1
102.7
107.4
112.3
117.2
122.1
127.2
132.3
137.6
142.9
148.3
500
122.7
128.4
134.3
140.3
146.5
152.7
159.0
165.4
172.0
178.6
185.3
1.33
1.35
1.37
1.39
1.41
1.44
1.47
1.50
1.52
1.56
1.59
FACTOR
To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY
Table 2
175
To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR 'NUMBER
FROM LONG. DIFF.
LONG.
DIFF.
MIDDLE LATITUDE
DEP.
41°
42°
43°
44°
45°
46°-
47°
48°
49°
50°
51°
51
12.5
13.1
13.7
14.3
14.9
15.6
16.2
16.9
17,5
18.2
18.9
52
12.8
13.4
14.0
14.6
15.2
15.9
16.5
17.2
17.9
18.6
19.3
53
13.0
13.6
14.2
14.9
15.5
16.2
16.9
17.5
18.2
18.9
19.6
54
13.2
13.9
14.5
15.2
15.8
16.5
17.2
17.9
18.6
19.3
20.0
55
13.5
14.1
14,8
15.4
16.1
16.8
17.5
18.2
18.9
19.6
20.4
56
13.7
14.4
lo.O
"15.7
16.4
17.1
17.8
18.5
19.3
20.0
20.8
57
14.0
14.6
15.3
16.0
16.7
17.4
18.1
18.9
19.6
20.4
21.1
58
14.2
14.9
15.6
16.3
17.0
17.7
18.4
19.2
19.9
20.7
21.5
59
14.5
15.2
15.9
16.6
17.3
18.0
18.8
19.5
20.3
21.1
21.9
60
14.7
15.4
16.1
16.8
17.6
18.3
19.1
19.9
20.6
21.4
22.2
61
15.0
15.7
16.4
17.1
17.9
18.6
19.4
20.2
21.0
21.8
22.6
62
15 9
15.9
16.7
17.4
18.2
18 9
19 7
20 5
21.3
22.1
?3 0
63
15.5
16.2
16.9
17.7
18.5
19.2
20.0
20.8
21.7
22.5
23.4
64
15.7
16.4
17.2
18.0
18.7
19.5
20.4
21.2
22.0
22.9
23.7
65
15.9
16.7
17.5
18.2
19.0
19.8
20.7
21.5
22.4
23.2
24.1
66
16.2
17.0
17.7
18.5
19.3
20.2
21.0
21.8
22.7
23.6
24.5
67
164
17.2
18.0
18.8
19.6
20 5
21.3
22.2
23.0
23.9
?4 8
68
16.7
17.5
18.3
19.1
19.9
20.8
21.6
22.5
23.4
24.3
25.2
69
16.9
17.7
18.5
19.4
20.2
21.1
21.9
22.8
23.7
24.6
25.6
70
17 ?
18.0
18.8
19.6
20.5
21 4
22.3
23.2
24.1
25.0
?5 9
71
17.4
18.2
19.1
19.9
20.8
21.7
22.6
23.5
24.4
25.4
26.3
72
177
18.5
19.3
20.2
21.1
22 0
22.9
23.8
24.8
25.7
?67
73
17.9
18.8
19.6
20.5
21.4
22.3
23.2
24.2
25.1
26.1
27.1
74
18.2
19.0
19.9
20.8
21.7
22.6
23.5
24.5
25.5
26.4
27.4
75
18.4
19.3
20.1
21.0
22.0
22.9
23.9
24.8
25.8
26.8
27.8
76
18.6
19.5
20.4
21.3
22.3
23.2
24.2
25.1
26.1
27.1
28.2
77
18 9
19 8
20 7
21.6
22 6
23 5
24.5
25.5
26.5
27.5
?85
78
19.1
20.0
21.0
21.9
22.8
23.8
24.8
25.8
26.8
27.9
28.9
79
19.4
20.3
21.2
22.2
23.1
24.1
25.1
26.1
27.2
28.2
29.3
80
19.6
20.5
21.5
22.5
23.4
24.4
25'.4
26.5
27.5
28.6
29.7
81
19.9
20.8
21.8
22.7
23.7
24.7
25.8
26.8
27.9
28.9
30.0
82
20.1
21.1
22.0
23.0
24.0
25.0
26.1
27.1
28.2
29.3
30.4
83
20.4
21.3
22.3
23.3
24.3
25.3
26.4
27.5
28.5
29.6
30.8
84
20.6
21.6
22.6
23.6
24.6
25.6
26.7
27.8
28.9
30.0
31.1
85
20.8
21.8
22.8
23.9
24.9
26.0
27.0
28.1
29.2
30.4
31.5
86
21.1
22.1
23.1
24.1
25.2
26.3
27.3
28.5
29.6
30.7
31.9
87
21.3
22.3
23.4
24.4
25.5
26.6
27.3;
28.8
29.9
31.1
32.2
88
21.6
22.6
23.6
24.7
25.8
26.9
28.0
29.1
30.3
31.4
32.6
89
21.8
22.9
23.9
25.0
26.1
27.2
28.3
29.4
30.6
31.8
33.0
90
22.1
23.1
24.2
25.3
26.4
27.5
28.6
29.8
31.0
32.1
33.4
91
22.3
23.4
24.4
25.5
26.7
27.8
28.9
30.1
31.3
32.5
33.7
92
22.6
23.6
24.7
25.8
26.9
28.1
29.3
30.4
31.6
32.9
34.1
93
22.8
23.9
25.0
26.1
27.2
28.4
29.6
30.8
32.0
33.2
34.5
94
23.1
24.1
25.3
26.4
27.5
28.7
29.9
31.1
32.3
33.6
34.8
95
23.3
24.4
25.5
26.7
27.8
29.0
30.2
31.4
32.7
33.9
35.2
96
23.5
24.7
25.8
26.9
28.1
29.3
30.5
31.8
33.0
34.3
35.6
97
23.8
24.9
26.1
27.2
28.4
29.6
30.8
32.1
33.4
34.6
36.0
98
24.0
25.2
26.3
27.5
28.7
29.9
31.2
32.4
33.7
35.0
36.3
99
24.3
25.4
26.6
27.8
29.0
30.2
31.5
32.8
34.1
35.4
36.7
100
24.5
25.7
26.9
28.1
29.3
30.5
31.8
33.1
34.4
35.7
37.1
600
147.2
154.1
161.2
168.4
175.7
183.2
190.8
198.5
206U
214.3
222.4
700
171.7
179.8
188.1
196.5
205.0
213.7
222.6
231.6
240.8
250.0
259.4
800
196.1
205.4
214.9
224.6
234.3
244.2
254.4
264.7
275.2
285.8
296.5
900
220.8
231.2
241.8
252.7
263.7
274.8
286.2
297.8
309.7
321.5
333.7
1.33
1.35
1.37
1.39
1.41
1.44
1.47
1.50
1.52
1.56
1.59
FACTOR
To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP.
176
Table 2
To CHANGE LONG. DIFF. INTO DEP., SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIPF.
OK
MIDDLE LATITUDE
DBF.
52°
53°
54°
55°
56°
57°
58°
59°
60°
1
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
2
0.8
0.8
0.8
0.9
0.9
0.9
0.9
1.0
1.0
3
1.2
1.2
1.2
1.3
1.3
1.4
1.4
1.5
1.5
4
1.5
1.6
1.6
1.7
1.8
1.8
1.9
1.9
2.0
5
1.9
2.0
2.1
2.1
2.2
2.3
2.4
2.4
2.5
6
2.3
2.4
2.5
2.6
2.6
2.7
2.8
2.9
3.0
7
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
8
3.1
3.2
3.3
3.4
3.5
3.6
3.8
3.9
4.0
9
3.5
3.6
3.7
3.8
4.0
4.1
4.2
4.4
4.5
10
3.8
4.0
4.1
4.3
4.4
4.6
4.7
4.8
5.0
11
4.2
4.4
4.5
4.7
4.8
5.0
5.2
5.3
5.5
12
4.6
4.8
4.9
5.1
5.3
5.5
5.6
5.8
6.0
13
5.0
5.2
5.4
5.5
5.7
5.9
6.1
6.3
6.5
14
5.4
5.6
5.8
6.0
6.2
6.4
6.6
6.8
7.0
15
5.8
6.0
6.2
6.4
6.6
6.8
7.1
7.3
7.5
16
6.1
6.4
6.6
6.8
7.1
7.3
7.5
7.8
8.0
17
6.5
6.8
7.0
7.2
7.5
7.7
8.0
8.2
8.5
18
6.9
7.2
7.4
7.7
7.9
8.2
8.5
8.7
9.0
19
7.3
7.6
7.8
8.1
8.4
8.7
8.9
9.2
9.5
20
7.7
8.0
8.2
8.5
8.8
9.1
9.4
9.7
10.0
21
8.1
8.4
8.7
9.0
9.3
9.6
9.9
10.2
10.5
22
8.5
8.8
9.1
9.4
9.7
10.0
10.3
10.7
11.0
23
8.8
9.2
9.5
9.8
10.1
10.5
10.8
11.2
11.5
24
9.2
9.6
9.9
10.2'
10.6
10.9
11.3
11.6
12.0
25
9.6
10.0
10.3
10.7
11.0
11.4
11.8
12.1
12.5
26
10.0
10.4
10.7
11.1
11.5
11.8
12.2
12.6
13.0
27
10.4
10.8
11.1
11.5
11.9
12.3
12.7
13.1
13.5
28
10.8
11.1
11.5
11.9
12.3
12.8
13.2
13.6
14.0
29
11.1
11.5
12.0
12.4
12.8
13.2
13.6
14.1
14.5
30
11.5
11.9
12.4
12.8
13.2
13.7
14.1
14.5
15.0
31
11.9
12.3
12.8
13.2
13.7
14.1
14.6
15.0
15.5
32
12.3
12.7
13.2
13.6
14.1
14.6
15.0
15.5
16.0
33
12.7
13.1
13.6
14.1
14.5
15.0
15.5
16.0
16.5
34
13.1
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
35
13.5
13.9
14.4
14.9
15.4
15.9
16.5
17.0
17.5
36
13.8
14.3
14.8
15.4
15.9
16.4
16.9
17.5
18.0
37
14.2
14.7
15.3
15.8
16.3
16.8
17.4
17.9
18.5
38
14.6
15.1
15.7
16.2
16.8
17.3
17.9
18.4
19.0
39
15.0
15.5
16.1
16.6
17.2
17.8
18.3
18.9
19.5
40
15.4
15.9
16.5
•17.1
17.6
18.2
18.8
19.4
20.0
41
15.8
16.3
16.9
17.5
18.1
18.7
19.3
19.9
20.5
42
16.1
16.7
17.3
17.9
18.5
19.1
19.7
20.4
21.0
43
16.5
17.1
17.7
18.3
19.0
19.6
20.2
209
21.5
44
16.9
17.5
18.1
18.8
19.4
20.0
20.7
21.3
22.0
45
17.3
17.9
18.5
19.2
19.8
20.5
21.2
21.8
22.5
46
17.7
18.3
19.0
19.6
20.3
20.9
21.6
22.3
23.0
47
18.1
18.7
19.4
20.0
20.7
21.4
22.1
22.8
23.5
48
18.4
19.1
19.8
20.5
21.2
21.9
22.6
23.3
24.0
49
18.8
19.5
20.2
20.9
21.6
22.3
23.0
23.8
24.5
50
19.2
19.9
20.6
21.3
22.0
22.8
23.5
24.2
25.0
100
38.4
39.8
41.2
42.6
44.1
45.5
47.0
48.5
50.0
200
76.9
79.6
82.4
85.3
88.2
91.1
94.0
97.0
100.0
300
115.3
119.5
123.7
127.9
132.2
136.6
141.0
145.5
150.0
400
153.7
159.3
164.9
170.6
176.3
182.2
188.1
194.0
200.0
500
192.2
199.1
206.1
213.2
220.4
227.7
235.0
242.5
250.0
1.62
1.66
1.70
1.74
1.79
1.84
1.89
1.94
2.00
FACTOR
To CHANGE DEP. INTO LONG. DIFF., MULTIPLY TABULAR NUMBER BY
Table 2
177
To CHANGE LONG. DIFF. INTO DEP. SUBTRACT TABULAR NUMBER
FROM LONG. DIFF.
LONG.
DIFF.
MIDDLE LATITUDE
OR
DEP.
52°
53°
54°
55°
56°
57°
58°
59°
60°
51
19.6
20.3
21.0
21.7
22.5
23.2
24.0
24.7
25.5
52
20.0
20.7
21.4
22.2
22.9
23.7
24.4
25.2
26.0
53
20.4
21.1
21.8
22.6
23.4
24.1
24.9
25.7
26.5
54
20.8
21.5
22.3
23.0
23.8
24.6
25.4
26.2
27.0
55
21.1
21.9
22.7
23.5
24.2
25.0
25.9
26.7
27.5
56
21.5
22.3
23.1
23.9
24.7
25.5
26.3
27.2
28.0
57
21.9
22.7
23.5
24.3
25.1
26.0
26.8
27.6
28.5
58
22.3
23.1
23.9
24.7
25.6
26.4
27.3
28.1
29.0
59
22.7
23.5
24.3
25.2
26.0
26.9
27.7
28.6
29.5
60
23.1
23.9
24.7
25.6
26.4
27.3
28.2
29.1
30.0
61
23.4
24.3
25.1
26.0
26.9
27.8
28.7
29.6
30.5
62
23.8
24.7
25.6
26.4
27.3
28.2
29.1
30.1
31.0
63
24.2
25.1
26.0
26.9
27.8
28.7
29.6
30.6
31.5
64
24.6
25.5
26.4
27.3
28.2
29.1
30.1
31.0
32.0
65
25.0
25.9
26.8
27.7
28.7
29.6
30.6
31.5
32.5
66
25.4
26.3
27.2
28.1
29.1
30.1
31.0
32.0
33.0
67
25.8
26.7
27.6
28.6
29.5
30.5
31.5
32.5
33.5
68
26.1
27.1
28.0
29.0
30.0
31.0
32.0
33.0
34.0
69
26.5
27.5
28.4
29.4
30.4
31.4
32.4
33.5
34.5
70
26.9
27.9
28.9
29.8
30.9
31.9
32.9
33.9
35.0
71
27.3
28.3
29.3
30.3
31.3
32.3
33.4
34.4
35.5
72
27.7
28.7
29.7
30.7
31.7
32.8
33.8
34.9
36.0
73
28.1
29.1
30.1
31.1
32.2
33.2
34.3
35.4
36.5
74
28.4
29.5
30.5
31.6
32.6
33.7
34. S
35.9
37.0
75
28.8
29.9
30.9
32.0
33.1
34.2
35.3
36.4
37.5
76
29.2
30.3
31.3
32.4
33.5
34.6
35.7
36.9
38.0
77
29.6
30.7
31.7
32.8
33.9
35.1
36.2
37.3
38.5
78
30.0
31.1
32.2
33.3
34.4
35.5
36.7
37.8
39.0
79
30.4
31.5
32.6
33.7
34.8
36.0
37.1
38.3
39.5
80
30.7
31.9
33.0
34.1
35.3
36.4
37.6
38.8
40.0
81
31.1
32.3
33.4
34.5
35.7
36.9
38.1
39.3
40.5
82
31.5
32.7
33.8
35.0
36.1
37.3
38.5
39.8
41.0
83
31.9
33.0
34.2
35.4
36.6
37.8
39.0
40.3
41.5
84
32.3
33.4
34.6
35.8
37.0
38.3
39.5
40.7
42.0
85
32.7
33.8
35.0
36.2
37.5
38.7
40.0
41.2
42.5
86
33.1
34.2
35.5
36.7
37.9
39.2
40.4
41.7
43.0
87
33.4
34.6
35.9
37.1
38.4
39.6
40.9
42.2
43.5
88
33.8
35.0
36.3
37.5
38.8
40.1
41.4
42.7
44.0
89
34.2
35.4
36.7
38.0
39.2
40.5
41.8
43.2
44.5
90
34.6
35.8
37.1
38.4
39.7-
41.0
42.3
43.6
45.0
91
35.0
36.2
37.5
38.8
40.1
41.4
42.8
44.1
45.5
92
35.4
36.6
37.9
39.2
40.6
41.9
43.2
44.6
46.0
93
35.7
37.0
38.3
39.7
41.0
42.3
43.7
45.1
46.5
94
36.1
37.4
38.7
40.1
41.4
42.8
44.2
45.6
47.0
95
36.5
37.8
39.2
40.5
41.9
43.3
44.7
46.1
47.5
96
36.9
38.2
39.6
40.9
42.3
43.7
45.1
46.6
48.0
97
37.3
38.6
40.0
41.4
42.8
44.2
45.6
47.0
48.5
98
37.7
39.0
40.4
41.8
43.2
44.6
46.1
47.5
49.0
90
38.0
39.4
40.8
42.2
43.6
45.1
46.5
48.0
49.5
100
38.4
39,8
41.2
42.6
44.1
45.5
47.0
48.5
50.0
600
230.6
238.9
247.3
255.9
264.5
273.2
282.0
291.0
300.0
700
269.2
279.7
288.6
298.5
308.6
318.7
329.0
339.6
350.0
800
307.5
319.5
329.8
341.2
352.6
364.3
376.1
388.0
400.0
900
346.0
358.3
371.1
383.8
396.8
409.9
423.2
436.6
450.0
1.63
1.66
1.70
1.74
1.79
1.84
1.89
1.94
2.00
FACTOR
To CHANGE DEP. INTO LONG. DIFF. MULTIPLY TABULAR NUMBER BY
FACTOR AT FOOT OF COLUMN AND ADD PRODUCT TO DEP.
178
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
100
00000
043
087
130
173
217
260
303
346
389
01
432
475
518
561
604
647
689
732
775
817
44
43
42
02
860
903
945
988
*030
*072
*115
*157
*199
*242
1
4.4
4.3
4.2
03
01284
326
368
410
452
494
536
578
620
662
2
8.8
8.6
8.4
04
703
745
787
828
870
912
953
995
*036
*078
3
13.2
12.9
12.6
05
02119
160
202
243
284
325
366
407
449
490
4
17.6
17.2
16.8
06
531
572
612
653
694
735
776
816
857
898
5
22.0
21.5
21.0
6
26.4
25.8
25.2
07
938
979
*019
*060
*100
*141
*181
*222
*262
*302
7
30.8
30.1
29.4
08
03342
383
423
463
503
543
583
623
663
703
8
35.2
34.4
33.6
09
743
782
822
862
902
941
981
*021
*060
*100
9
39.6
38.7
37.8
110
04139
179
218
258
297
336
376
415
454
493
11
532
571
610
650
689
727
766
805
844
883
41
40
39
12
922
961
999
*038
*077
*115
*154
*192
*231
*269
1
4.1
4.0
3.9
13
05308
346
385
423
461
500
538
576
614
652
2
8.2
8.0
7.8
14
690
729
767
805
843
881
918
956
994
*032
3
12.3
12.0
11.7
15
06070
108
145
183
221
258
296
333
371
408
4
16.4
16.0
15.6
16
446
483
521
558
595
633
670
707
744
781
5
20.5
20.0
19.5
(>
24.6
24.0
23.4
17
819
856
893
930
967
*004
*041
*078
*115
*151
7
28.7
28.0
27.3
18
07188
225
262
298
335
372
408
445
482
518
8
32.8
32.0
31.2
19
555
591
628
664
700
737
773
809
846
882
9
36.9
36.0
35.1
120
918
954
990
*027
*063
*099
*135
*171
*207
*243
21
08279
314
350
386
422
458
493
529
565
600
38
37
36
22
636
672
707
743
778
814
849
884
920
955
1
3.8
3.7
3.6
23
991
*026
*061
*096
*132
*167
*202
*237
*272
*307
2
7.6
7.4
7.2
24
09342
377
412
447
482
517
552
587
621
656
3
11.4
11.1
10.8
25
26
691
10037
726
072
760
106
795
140
830
175
864
209
899
243
934
278
968
312
*003
346
4
5
6
15.2
19.0
22.8
14.8
18.5
22.2
14.4
18.0
21.6
27
380
415
449
483
517
551
585
619
653
687
7
26.6
25.9
25.2
28
721
755
789
823
857
890
924
958
992
*025
8
30.4
29.6
28.8
29
11059
093
126
160
193
227
261
294
327
361
9
34.2
33.3
32.4
130
394
428
461
494
528
561
594
628
661
694
31
727
760
793
826
860
893
926
959
992
*024
35
34
33
32
12057
090
123
156
189
222
254
287
320
352
1
3.5
3.4
3.3
33
385
418
450
483
516
548
581
613
646
678
2
7.0
6.8
6.6
34
35
36
710
13033
354
743
066
386
775
098
418
808
130
450
840
162
481
872
194
513
905
226
545
937
258
577
969
290
609
*001
322
640
3
4
5
6
10.5
14.0
17.5
21.0
10.2
13.6
17.0
20.4
9.9
13.2
16.5
19.8
37
672
704
735
767
799
830
862
893
925
956
7
24.5
23.8
23.1
38
988
*019
*051
*082
*114
»145
*176
*208
*239
*270
8
28.0
27.2
26.4
39
14301
333
364
395
426
457
489
520
551
582
9
31.5
30.6
29.7
140
613
644
675
706
737
768
799
829
860
891
41
922
953
983
*014
*045
*076
*106
*137
*168
*198
32
31
30
42
15229
259
290
320
351
381
412
442
473
503
1
3.2
3.1
3.0
43
534
564
594
625
655
685
715
746
776
806
2
6.4
6.2
6.0
44
45
46
836
16137
435
866
167
465
897
197
495
927
227
524
957
256
554
987
286
584
*017
316
613
*047
346
643
*077
376
673
*107
406
702
3
4
5
6
9.6
12.8
16.0
19.2
9.3
12.4
15.5
18.6
9.0
12.0
15.0
18.0
47
732
761
791
820
850
879
909
938
967
997
7
22.4
21.7
21.0
48
17026
056
085
114
143
173
202
231
260
289
8
25.6
24.8
24.0
49
319
348
377
406
435
464
493
522
551
580
9
28.8
27.9
27.0
150
609
638
667
696
725
754
782
811
840
869
0
1
2
3
4
5
6
7
8
9
Prop. Pta.
Table 3. Number Logarithms
179
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
150
17609
638
667
696
725
754
782
811
840
869
51
898
926
955
984
*013
*041
«070
«099
*127
*156
52
18184
213
241
270
298
327
355
384
412
441
53
469
498
526
554
583
611
639
667
696
724
54
752
780
808
837
865
893
921
949
977
*005
55
19033
061
089
117
145
173
201
229
257
285
56
312
340
368
396
424
451
479
507
535
562
57
590
618
645
673
700
728
756
783
811
838
58
866
893
921
948
976
«003
*030
*058
«085
*112
59
20140
167
194
222
249
276
303
330
358
385
160
412
439
466
493
520
548
575
602
629
656
61
683
710
737
763
790
817
844
871
898
925
29
28
27
62
952
978
*005
«032
«059
*085
*112
*139
*165
*192
1
2.9
2.8
2.7
63
21219
245
272
299
325
352
378
405
431
458
2
5.8
5.6
5.4
64
65
66
484
748
22011
511
775
037
537
801
063
564
827
089
590
854
115
617
880
141
643
906
167
669
932
194
696
958
220
722
985
246
3
4
5
8.7
11.6
14.5
8.4
11.2
14.0
8.1
10.8
13.5
6
17.4
16.8
16.2
67
272
298
324
350
376
401
427
453
479
505
7
20.3
19.6
18.9
68
531
557
583
608
634
660
686
712
737
763
8
23.2
22.4
21.6
69
789
814
840
866
891
917
943
968
994
*019
9
26.1
25.2
24.3
170
23045
070
096
121
147
172
198
223
249
274
71
300
325
350
376
401
426
452
477
502
528
.26
25
24
72
553
578
603
629
654
679
704
729
754
779
1
2.6
2.5
2.4
73
805
830
855
880
905
930
955
980
*005
*030
2
5.2
5.0
4.8
74
24055
080
105
130
155
180
204
229
254
279
3
7.8
7.5
7.2
75
304
329
353
378
403
428
452
477
502
527
4
10.4
10.0
9.6
76
551
576
601
625
650
674
699
724
748
773
5
13.0
12.5
12.0
6
15.6
15.0
14.4
77
797
822
846
871
895
920
944
969
993
*018
7
18.2
17.5
16.8
78
25042
066
091
115
139
164
188
212
237
261
8
20.8
20.0
19.2
79
285
310
334
358
382
406
431
455
479
503
9
23.4
22.5
21.6
180
527
551
575
600
624
648
672
696
720
744
81
768
792
816
840
864
888
912
935
959
983
23
22
21
82
26007
031
055
079
102
126
150
174
198
221
1
2.3
2.2
2.1
83
245
269
293
316
340
364
387
411
435
458
2
4.4
4.2
84
482
505
529
553
576
600
623
647
670
694
3
6^9
6.6
6.3
85
717
741
764
788
811
834
858
881
905
928
4
9.2
8.8
8.4
86
951
975
998
*021
*045
*068
*091
*114
*138
*161
5
11.5
11.0
10.5
6
13.8
13.2
12.6
87
27184
207
231
254
277
300
323
346
370
393
7
16.1
15.4
14.7
88
416
439
462
485
508
531
554
577
600
623
8
18.4
17.6
16.8
89
646
669
692
715
738
761
784
807
830
852
9
20.7
19.8
18.9
190
875
898
921
944
967
989
*012
*035
*058
«081
91
28103
126
149
171
194
217
240
262
285
307
92
330
353
375
398
421
443
466
488
511
533
93
556
578
601
623
646
668
691
713
735
758
94
780
803
825
847
870
892
914
937
959
981
95
29003
026
048
070
092
115
137
159
181
203
96
226
248
270
292
314
336
358
380
403
425
97
447
469
491
513
535
557
579
601
623
645
98
667
688
710
732
754
776
798
820
842
863
99
885
907
929
951
973
994
«016
*038
*060
*081
200
30103
125
146
168
190
211
233
255
276
298
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
180
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
200
30103
125
146
168
190
211
233
255
276
2<)8
01
320
341
363
384
406
428
449
471
492
514
02
535
557
578
600
621
643
664
685
707
728
03
750
771
792
814
835
856
878
899
920
942
04
963
984
*006
«027
*048
*069
*091
*112
*133
*154
05
31175
197
218
239
260
281
302
323
345
366
06
387
408
429
450
471
492
513
534
555
576
07
597
618
639
660
681
702
723
744
765
785
08
806
827
848
869
890
911
931
952
973
994
09
32015
035
056
077
098
118
139
160
181
201
210
222
243
263
284
305
325
346
366
387
408
11
428
449
469
490
510
531
552
572
593
613
22
21
20
12
634
654
675
695
715
736
756
777
797
818
1
2.2
2.1
2.0
13
838
858
879
899
919
940
960
980
*001
*021
2
4.4
4.2
4.0
3
6.6
6.3
6.0
14
33041
062
082
102
122
143
163
183
203
224
4
8.8
8.4
8.0
15
244
264
284
304
325
345
365
385
405
425
5
11.0
10.5
10.0
16
445
465
486
506
526
546
566
586
606
626
6
13.2
12.6
12.0
17
18
19
646
846
34044
666
866
064
686
885
084
706
905
104
726
925
124
746
945
143
766
965
163
786
985
183
806
*005
203
826
*025
223
7
8
9
15.4
17.6
19.8
14.7
16.8
18.9
14.0
16.0
18.0
220
242
262
282
301
321
341
361
380
400
420
21
439
459
479
498
518
537
557
577
596
616
22
635
655
674
694
713
733
753
772
792
811
23
830
850
869
889
908
928
947
967
986
*005
24
35025
044
064
083
102
122
141
160
180
199
25
218
238
257
276
295
315
334
353
372
392
26
411
430
449
468
488
507
526
545
564
583
27
603
622
641
660
679
698
717
736
755
774
28
793
813
832
851
870
889
908
927
946
965
29
984
*003
*021
*040
*059
*078
*097
*116
*135
*154
230
36173
192
211
229
248
267
286
305
324
342
31
361
380
399
418
436
455
474
493
511
530
19
18
17
32
549
568
586
605
624
642
661
680
698
717
1
1.9
1.8
1.7
33
736
754
773
791
810
829
847
866
884
903
2
3.8
3.6
3.4
3
5.7
5.4
5.1
34
922
940
959
977
996
*014
*033
*051
*070
*088
4
7^6
7.2
6.8
35
37107
125
144
162
181
199
218
236
254
273
5
9.5
9.0
8^5
36
291
310
328
346
365
383
401
420
438
457
6
11A
10.8
10.2
37
475
493
511
530
548
566
585
603
621
639
7
13.3
12.6
11.9
38
658
676
694
712
731
749
767
785
803
822
8
15.2
14.4
13.6
39
840
858
876
894
912
931
949
967
985
*003
9
17.1
16.2
15.3
240
38021
039
057
075
093
112
130
148
166
184
41
202
220
238
256
274
292
310
328
346
364
42
382
399
417
435
453
471
489
507
525
543
43
561
578
596
614
632
650
668
686
703
721
44
739
757
775
792
810
828
846
863
881
899
45
917
934
952
970
987
*005
*023
*041
*058
*076
46
39094
111
129
146
164
182
199
217
235
252
47
270
287
305
322
340
358
375
393
410
428
48
445
463
480
498
515
533
550
568
585
602
49
620
637
655
672
690
707
724
742
759
777
250
794
811
829
846
863
881
898
915
933
950
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
181
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
250
39794
811
829
846
863
881
898
915
933
950
51
967
985
«002
*019
*037
*054
*071
*088
*106
*123
52
40140
157
175
192
209
226
243
261
278
295
53
312
329
346
364
381
398
415
432
449
466
54
483
500
518
535
552
569
586
603
620
637
55
654
671
688
705
722
739
756
773
790
807
56
824
841
858
875
892
909
926
943
960
976
57
993
*010
«027
«044
*061
*078
*095
*111
*128
*145
58
41102
179
196
212
229
246
263
280
296
313
59
3:30
347
303
380
397
414
430
447
464
481
260
497
514
531
547
564
581
597
614
631
647
61
664
681
697
714
731
747
764
780
797
814
18
17 16
62
830
847
863
880
896
913
929
946
963
979
1 1.8
1
.7 1.6
63
996
*012
*029
*045
*062
«078
*095
*111
*127
*144
2 3.6
3.4 3.2
64
65
66
42160
325
488
177
341
504
193
357
521
210
374
537
226
390
553
243
406
570
259
423
586
275
439
602
292
455
619
308
472
635
3 5.4
4 7.2
5 9.0
6 10.8
5.1 4.8
6.8 6.4
8.5 8.0
10.2 9.6
67
651
667
684
700
716
732
749
765
781
797
7 12.6
11.9 11.2
68
813
830
846
862
878
894
911
927
943
959
8 14.4
13.6 12.8
69
975
991
*008
*024
«040
*056
«072
*088
*104
*120
9 16.2
15.3 14.4
270
43136
152
169
185
201
217
233
249
265
281
71
297
313
329
345
361
377
393
409
425
441
72
457
473
489
505
521
537
553
569
584
600
73
• 616
632
648
664
680
696
712
727
743
759
74
775
791
807
823
838
854
870
886
902
917
75
933
949
965
981
996
*012
*028
*044
*059
*075
76
44091
107
122
138
154
170
185
201
217
232
77
248
264
279
295
311
326
342
358
373
389
78
404
420
436
451
467
483
498
514
529
545
79
5<>0
576
592
607
623
638
654
(569
685
700
280
716
731
747
762
778
793
809
824
840
855
81
871
886
902
917
932
948
963
979
994
*010
15
14
82
45025
040
056
071
086
102
117
133
148
163
1 1
5
1.4
83
179
194
209
225
240
255
271
286
301
317
2 3
2.8
84
332
347
362
378
393
408
423
439
454
469
3 4.5
4.2
85
484
500
515
530
545
561
576
.591
606
621
4 6.0
5.6
86
637
652
667
682
697
712
728
743
758
773
5 7.5
7.0
6 9
0
8.4
87
788
803
818
834
849
864
879
894
909
924
7 10.5
9.8
88
939
954
969
984
*000
*015
*030
*045
*060
*075
8 12.0
11.2
89
46090
105
120
135
150
165
180
195
210
225
9 13.5
12.6
290
240
255
270
285
300
315
330
345
359
374
91
389
404
419
434
449
464
479
494
509
523
92
538
553
568
583
598
613
627
642
657
672
93
687
702
716
731
746
761
776
790
805
820
94
835
850
864
879
894
909
923
938
953
967
95
982
997
*012
*026
*041
«056
*070
*085
*100
*114
96
47129
144
159
173
188
202
217
232
246
261
97
276
290
305
319
334
349
363
378
392
407
98
422
436
451
465
480
494
509
524
538
553
99
567
582
596
611
625
640
654
669
683
698
300
712
727
741
756
770
784
799
813
828
842
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
182
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
300
47712
727
741
756
770
784
799
813
828
842
01
857
871
885
900
914
929
943
958
972
986
02
48001
015
029
044
058
073
087
101
116
130
03
144
159
173
187
202
216
230
244
259
273
04
287
302
316
330
344
359
373
387
401
416
05
430
444
458
473
487
501
515
530
544
558
06
572
586
601
615
629
643
657
671
686
700
07
714
728
742
756
770
785
799
813
827
841
08
855
869
883
897
911
926
940
954
968
982
09
996
*010
*024
*038
*052
*066
*080
*094
*108
*122
310
49136
150
164
178
192
206
220
234
248
262
11
276
290
304
318
332
346
360
374
388
402
15
14
12
415
429
443
457
471
485
499
513
527
541
1
1.5
1.4
13
554
568
582
596
610
624
638
651
665
679
2
3.0
2.8
3
4.5
4.2
14
693
707
721
734
748
762
776
790
803
817
4
6.0
5.6
15
831
845
859
872
886
900
914
927
941
955
5
7.5
7.0
16
969
982
996
*010
*024
*037
*051
*065
*079
*092
6
9.0
8.4
17
50106
120
133
147
161
174
188
202
215
229
7
10.5
9.8
18
243
256
270
284
297
311
325
338
352
365
8
12.0
11.2
19
379
393
406
420
433
447
461
474
488
501
9
13.5
12.6
320
515
529
542
556
569
583
596
610
623
637
21
651
664
678
691
705
718
732
745
759
772
22
786
799
813
826
840
853
866
880
893
907
23
920
934
947
961
974
987
*001
*014
*028
*041
24
51055
068
081
095
108
121
135
148
162
175
25
188
202
215
228
242
255
268
282
295
308
26
322
335
348
362
375
388
402
415
428
441
27
455
468
481
495
508
521
534
548
561
574
28
587
601
614
627
640
654
667
680
693
706
29
720
733
746
759
772
786
799
812
825
838
330
851
865
878
891
904
917
930
943
957
970
31
983
996
*009
*022
*035
*048
*061
*075
*088
*101
13
12
32
52114
127
140
153
166
179
192
205
218
231
1
1.3
1.2
33
244
257
270
284
297
310
323
336
349
362
2
2.6
2.4
3
3.9
3.6
34
375
388
401
414
427
440
453
466
479
492
4
5.2
4.8
35
504
517
530
543
556
• 569
582
595
608
621
5
6.5
6.0
36
634
647
660
673
686
699
711
724
737
750
6
7.8
7.2
37
763
776
789
802
815
827
840
853
866
879
7
9.1
8.4
38
892
905
917
930
943
956
969
982
994
*007
8
10.4
9.6
39
53020
033
046
058
071
084
097
110
122
135
9
11.7
10.8
340
148
161
173
186
199
212
224
237
250
263
41
275
288
301
314
326
339
352
364
377
390
42
403
415
428
441
453
466
479
491
504
517
43
529
542
555
567
580
593
605
618
631
643
44
656
668
681
694
706
719
732
744
757
769
45
782
794
807
820
832
845
857
870
882
895
46
908
920
933
945
958
970
983
995
*008
*020
47
54033
045
058
070
083
095
108
120
133
145
48
158
170
183
195
208
220
233
245
258
270
49
283
295
307
320
332
345
357
370
382
394
350
407
419
432
444
456
469
481
494
506
518
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
183
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
350
54407
419
432
444
456
469
481
494
506
518
51
531
543
555
568
580
593
605
617
630
642
52
654
667
679
691
704
716
728
741
753
765
53
777
790
802
814
827
839
851
864
876
888
54
900
913
925
937
949
962
974
986
998
*011
55
55023
035
047
060
072
084
096
108
121
133
56
145
157
169
182
194
206
218
230
242
255
57
267
279
291
303
315
328
340
352
364
376
58
388
400
413
425
437
449
461
473
485
497
59
509
522
534
546
558
570
582
594
606
618
360
630
642
654
666
678
691
703
715
727
739
61
751
763
775
787
799
811
823
835
847
859
13
12
62
871
883
895
907
919
931
943
955
967
979
1
1.3
1.2
63
991
*003
«015
*027
*038
*050
*062
*074
*086
*098
2
2.6
2.4
64
56110
122
134
146
158
170
182
194
205
217
3
3.9
3.6
65
229
241
253
265
277
289
301
312
324
336
4
5.2
4.8
66
348
360
372
384
396
407
419
431
443
455
5
6
6.5
7.8
6.0
7.2
67
467
478
490
502
514
526
538
549
561
573
7
9.1
8.4
68
585
597
608
620
632
644
656
667
679
691
8
10.4
9.6
69
703
714
726
738
750
761
773
785
797
808
9
11.7
10.8
370
820
832
844
855
867
879
891
902
914
926
71
937
949
961
972
984
996
*008
*019
*031
*043
72
57054
066
078
089
101
113
124
136
148
159
73
171
183
194
206
217
229
241
252
264
276
74
287
299
310
322
334
345
357
368
380
392
75
403
415
426
438
449
461
473
484
496
507
76
519
530
542
553
565
576
588
600
611
623
77
634
646
657
669
680
692
703
715
726
738
78
749
761
772
784
795
807
818
830
841
852
79
864
875
887
898
910
921
933
944
955
967
380
978
990
«001
*013
*024
*035
*047
«058
*070
*081
81
58092
104
115
127
138
149
161
172
184
195
11
10
82
206
218
229
240
252
263
274
286
297
309
1
1.1
1 0
83
320
331
343
354
365
377
388
399
410
422
2
JL*U
2.0
84
433
444
456
467
478
490
501
512
524
535
3
3.3
3.0
85
546
557
569
580
591
602
614
625
636
647
4
4.4
4.0
86
659
670
681
692
704
715
726
737
749
760
5
5.5
5.0
6
6.6
6.0
87
771
782
794
805
816
827
838
850
861
872
7
7.7
7.0
88
883
894
906
917
928
939
950
961
973
984
8
8.8
8.0
89
995
*006
«017
*028
*040
«051
*062
*073
*084
*095
9
9.9
9.0
390
59106
118
129
140
151
162
173
184
195
207
91
218
229
240
251
262
273
284
295
306
318
92
329
340
351
362
373
384
395
406
417
428
93
439
450
461
472
483
494
506
517
528
539
94
550
561
572
583
594
605
616
627
638
649
95
660
671
682
693
704
715
726
737
748
759
96
770
780
791
802
813
824
835
846
857
868
97
879
890
901
912
923
934
945
956
966
977
98
988
999
«010
*021
»032
*043
«054
*065
«076
*086
99
60097
108
119
130
141
152
163
173
184
195
400
206
217
228
239
249
260
271
282
293
304
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
184
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
400
60206
217
228
239
249
260
271
282
293
304
01
314
325
336
347
358
369
379
390
401
412
02
423
433
444
455
466
477
487
498
509
520
03
531
541
552
5(53
574
584
595
606
617
627
04
638
649
660
670
681
692
703
713
724
735
05
746
756
767
778
788
799
810
821
831
842
06
853
863
874
885
895
906
917
927
938
949
07
959
970
981
991
*002
*013
*023
*034
*045
*055
08
61066
077
087
098
109
119
130
140
151
162
09
172
183
194
204
215
225
236
247
257
268
410
278
289
300
310
321
331
342
352
363
374
11
384
395
405
416
426
437
448
458
469
479
12
490
500
511
521
532
542
553
563
574
584
13
595
606
616
627
637
648
658
669
679
690
14
700
711
721
731
742
752
763
773
784
794
15
805
815
826
836
847
857
868
878
888
899
16
909
920
930
941
951
962
972
982
993
*003
17
62014
024
034
045
055
066
076
086
097
107
18
118
128
138
149
159
170
180
190
201
211
19
221
232
242
252
263
273
284
294
304
315
420
325
335
346
356
366
377
387
397
408
418
21
428
439
449
459
409
480
490
500
511
521
11 10 9
22
531
542
552
562
572
583
593
603
613
624
1 1.1 1.0 0.9
23
634
644
655
<>65
675
685
696
706
716
726
2 2.2 2.0 1.8
24
737
747
757
767
778
788
798
808
818
829
3 3.3 3.0 2.7
25
839
849
859
870
880
890
900
910
921
931
4 4.4 4.0 3.6
26
941
951
961
972
982
992
*002
*012
*022
*033
5 5.5 5.0 4.5
6 6.6 6.0 5.4
27
63043
053
063
073
083
094
104
114
124
134
7 7.7 7.0 6.3
28
144
155
165
175
185
195
205
215
225
236
8 8.8 8.0 7.2
29
246
256
266
276
286
296
306
317
327
337
9 9.9 9.0 8.1
430
347
357
367
377
387
397
407
417
428
438
31
448
458
468
478
488
498
508
518
528
538
32
548
558
568
579
589
599
609
619
629
639
33
649
659
669
679
689
699
709
719
729
739
34
749
759
769
779
789
799
809
819
829
839
35
849
859
869
879
889
899
909
919
929
939
36
949
959
969
979
988
998
*008
*018
*028
*038
37
64048
058
068
078
088
098
108
118
128
137
38
147
157
167
177
187
197
207
217
227
237
39
246
256
266
276
286
296
306
316
326
335
440
345
355
365
375
385
395
404
414
424
434
41
444
454
464
473
483
493
503
513
523
532
42
542
552
562
572
582
591
601
611
621
631
43
640
650
660
670
680
689
699
709
719
729
44
738
748
758
768
777
787
797
807
816
826
45
836
846
856
865
875
885
895
904
914
924
46
933
943
953
963
972
982
992
*002
*011
*021
47
65031
040
050
060
070
079
089
099
108
118
48
128
137
147
157
167
176
186
196
205
215
49
225
234
244
254
263
273
283
292
302
312
450
321
331
341
350
360
369
379
389
398
408
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
185
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
450
65 321
331
341
350
360
369
379
389
398
408
51
418
427
437
447
456
466
475
485
495
504
52
514
523
533
543
552
562
571
581
591
600
53
610
619
629
639
648
658
667
677
686
696
54
706
715
725
734
744
753
763
772
782
792
55
801
811
820
830
839
849
858
868
877
887
56
896
900
916
925
935
944
954
963
973
982
57
992
*001
*011
*020
*030
*039
*049
*058
*068
*077
58
66087
096
106
115
124
134
143
153
162
172
59
181
191
200
210
219
229
238
247
257
266
460
276
285
295
304
314
323
332
342
351
361
61
370
380
389
398
408
417
427
436
445
455
62
464
474
483
492
502
511
521
530
539
549
63
558
567
577
586
596
605
614
624
633
642
64
652
661
671
680
689
699
708
717
727
736
65
745
755
764
773
783
792
801
811
820
829
66
839
848
857
867
876
885
894
904
913
922
67
932
941
950
960
969
978
987
997
*006
*015
68
67 025
034
043
052
062
071
080
089
099
108
69
117
127
136
145
154
164
173
182
191
201
470
210
219
228
237
247
256
265
274
284
293
71
302
311
321
330
339
348
357
367
376
385
10 9 8
72
394
403
413
422
431
440
449
459
468
477
1 1.0 0.9 0.8
73
486
495
504
514
523
532
541
550
560
569
2 2.0 1.8 1.6
74
75
76
578
669
761
587
679
770
596
688
779
605
697
•788
614
706
797
624
715
806
633
724
815
642
733
825
651
742
834
660
752
843
3 3.0 2.7 2.4
4 4.0 3.6 3.2
5 5.0 4.5 4.0
6 6.0 5.4 4.8
77
852
861
870
879
888
897
906
916
925
934
7 7.0 6.3 5.6
78
943
952
961
970
979
988
997
*006
*015
*024
8 8.0 7.2 6.4
79
68034
043
052
061
070
079
088
097
106
115
9 9.0 8.1 7.2
480
124
133
142
151
160
169
178
187
196
205
81
215
224
233
242
251
260
269
278
287
296
82
305
314
323
332
341
350
359
368
377
386
83
395
404
413
422
431
440
449
458
467
476
84
485
494
502
511
520
529
538
547
556
565
85
574
583
592
601
610
619
628
637
646
655
86
664
673
681
690
699
708
717
726
735
744
87
753
762
771
780
789
797
806
815
824
833
88
842
851
860
869
878
886
895
904
913
922
89
931
940
949
958
9(56
975
984
993
*002
*011
490
69020
028
037
046
055
064
073
082
090
099
91
108
117
126
135
144
152
161
170
179
188
92
197
205
214
223
232
241
249
258
267
276
93
285
294
302
311
320
329
338
346
355
364
94
373
381
390
S99
408
417
425
434
443
452
95
461
469
478
487
496
504
513
522
531
539
96
548
557
566
574
583
592
601
609
618
627
97
636
644
653
662
671
679
688
697
705
714
98
723
732
740
749
758
767
775
784
793
801
99
810
819
827
836
845
854
862
871
880
888
500
897
906
914
923
932
940
949
958
966
975
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
186
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
500
69 897
906
914
923
932
940
949
958
966
975
01
984
992
*001
*010
*018
*027
*036
*044
*053
*062
02
70070
079
088
096
105
114
122
131
140
148
03
157
163
174
183
191
200
209
217
226
234
04
243
252
260
269
278
286
295
303
312
321
05
329
338
346
355
364
372
381
389
398
406
06
415
424
432
441
449
458
467
475
484
492
07
501
509
518
526
535
544
552
561
569
578
08
586
595
603
612
621
629
638
646
655
663
09
672
680
689
697
706
714
723
731
740
749
510
757
766
774
783
791
800
808
817
825
834
11
842
851
859
8(58
876
885
893
902
910
919
12
927
935
944
952
961
969
978
986
995
*003
13
71012
020
029
037
046
054
063
071
079
088
14
096
105
113
122
130
139
147
155
164
172
15
181
189
198
206
214
223
231
240
248
257
16
265
273
282
290
299
307
315
324
332
341
17
349
357
366
374
383
391
399
408
416
425
18
433
441
450
458
466
475
483
492
500
508
19
517
525
533
542
550
559
567
575
584
592
520
600
609
617
625
'634
642
650
659
667
675
21
684
692
700
709
717
725
734
742
750
759
987
22
767
775
784
792
800
809
817
825
834
842
1 0.9 0.8 0.7
23
850
858
867
875
883
892
900
908
917
925
2 1.8 1.6 1.4
24
933
941
950
958
966
975
983
991
999
*008
3 2.7 2.4 2.1
25
72016
024
032
041
049
057
066
074
082
090
4 3.6 3.2 2.8
26
099
107
115
123
132
140
148
156
165
173
5 4.5 4.0 3.5
6 5.4 4.8 4.2
27
181
189
198
206
214
222
230
239
247
255
7 6.3 5.6 4.9
28
263
272
280
288
296
304
313
321
329
337
8 7.2 6.4 5.6
29
346
354
362
370
378
387
395
403
411
419
9 8.1 7.2 6.3
530
428
436
444
452
460
469
477
485
493
501
31
509
518
526
534
542
550
558
567
575
583
32
591
599
607
616
624
632
640
648
656
665
33
673
681
689
697
705
713
722
730
738
746
34
754
762
770
779
787
795
803
811
819
827
35
835
843
852
860
868
876
884
892
900
908
36
916
925
933
941
949
957
965
973
981
989
37
997
*006
«014
*022
*030
*038
*046
*054
*062
*070
38
73078
086
094
102
111
119
127
135
143
151
39
159
167'
175
183
191
199
207
215
223
231
540
239
247
255
263
272
280
288
296
304
312
41
320
328
336
344
352
360
368
376
384
392
42
400
408
416
424
432
440
448
456
464
472
43
480
488
496
504
512
520
528
536
544
552
44
560
568
576
584
592
600
608
616
624
632
45
640
648
656
664
672
679
687
695
703
711
46
719
727
735
743
751
759
767
775
783
791
47
799
807
815
823
830
838
846
854
862
870
48
878
886
894
902
910
918
926
933
941
949
49
957
965
973
981
989
997
*005
*013
*020
*028
550
74036
044
052
060
068
076
084
092
099
107
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
187
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
550
74036
044
052
060
068
076
084
092
099
107
51
115
123
131
139
147
155
162
170
178
186
52
194
202
210
218
225
233
241
249
257
265
53
273
280
288
296
304
312
320
327
335
343
54
351
359
367
374
382
390
398
406
414
421
55
429
437
445
453
461
468
476
484
492
500
56
507
515
523
531
539
547
554
562
570
578
57
586
593
601
609
617
624
632
640
648
656
58
663
671
679
687
695
702
710
718
726
733
59
741
749
757
764
772
780
788
796
803
811
560
819
827
834
842
850
858
865
873
881
889
61
896
904
912
920
927
935
943
950
958
966
62
974
981
989
997
*005
*012
*020
•028
*035
«043
63
75051
059
066
074
082
089
097
105
113
120
64
128
136
143
151
159
166
174
182
189
197
65
205
213
220
228
236
243
251
259
266
274
66
282
289
297
305
312
320
328
335
343
351
67
358
366
374
381
389
397
404
412
420
427
68
435
442
450
458
465
473
481
488
496
504
69
511
519
526
534
542
549
557
565
572
580
570
587
595
603
610
618
626
633
641
648
656
71
664
671
679
686
694
702
709
717
724
732
8 7
72
740
747
755
762
770
778
785
793
800
808
1 0.8 0.7
73
815
823
831
838
846
853
861
868
876
884
2 1.6 1.4
74
891
899
906
914
921
929
937
944
952
959
3 2.4 2.1
75
967
974
982
989
997
*005
*012
«020
*027
*035
4 3.2 2.8
76
76042
050
057
065
072
080
087
095
103
110
5 4.0 3.5
6 4.8 4.2
77
118
125
133
140
148
155
163
170
178
185
7 5.6 4.9
78
193
200
208
215
223
230
238
245
253
260
8 6.4 5.6
79
268
275
283
290
298
305
313
320
328
335
9 7.2 6.3
580
343
350
358
365
373
380
388
395
403
410
81
418
425
433
440
448
455
462
470
477
485
82
492
500
507
515
522
530
537
545
552
559
83
567
574
582
589
597
604
612
619
626
634
84
641
649
656
664
671
678
686
693
701
708
85
716
723
730
738
745
753
760
768
775
782
86
790
797
805
812
819
827
834
842
849
856
87
864
871
879
886
893
901
908
916
923
930
88
938
945
953
960
967
975
982
989
997
«004
89
77012
019
026
034
041
048
056
063
070
078
590
085
093
100
107
115
122
129
137
144
151
91
159
166
173
181
188
195
203
210
217
225
92
232
240
247
254
262
269
276
283
291
298
93
305
313
320
327
335
342
349
357
364
371
94
379
386
393
401
408
415
422
430
437
444
95
452
459
466
474
481
488
495
503
510
517
96
525
532
539
546
554
561
568
576
583
590
97
597
605
612
619
627
634
641
648
656
663
98
670
677
685
692
699
706
714
721
728
735
99
743
750
757
764
772
779
786
793
801
808
600
815
822
830
837
844
851
859
866
873
880
0
1
2
3
4
5
6
7
8
9
Prop. Pta.
188
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
600
77815
822
830
837
844
851
859
8(56
873
880
01
887
895
902
909
916
924
931
938
945
952
02
960
967
974
981
988
996
*003
*010
*017
*025
03
78032
039
046
053
061
068
075
082
089
097
04
104
111
118
125
132
140
147
154
161
168
05
176
183
190
197
204
211
219
226
233
240
06
247
254
262
269
276
283
290
297
305
312
07
319
326
333
340
347
355
362
369
376
383
08
390
398
405
412
419
426
433
440
447
455
09
462
469
476
483
490
497
504
512
519
526
610
533
540
547
554
561
569
576
583
590
597
11
604
611
618
625
633
640
647
654
661
668
12
675
682
689
696
704
711
718
725
732
739
13
746
753
760
767
774
781
789
796
803
810
14
817
824
831
838
845
852
859
866
873
880
15
888
895
902
909
916
923
930
937
944
951
16
958
965
972
979
986
993
*000
*007
*014
*021
17
79029
036
043
050
057
064
071
078
085
092
18
099
106
113
120
127
134
141
148
155
162
19
169
176
183
190
197
204
211
218
225
232
620
239
246
253
260
267
274
281
288
295
302
21
309
316
323
330
337
344
351
358
365
372
876
22
379
386
393
400
407
414
421
428
435
442
1 0.8 0.7 0.6
23
449
456
463
470
477
484
491
498
505
511
2 1.6 1.4 1.2
24
25
26
518
588
657
525
595
664
532
602
671
539
609
678
546
616
685
553
623
692
560
630
699
567
637
706
574
644
713
581
650
720
3 2.4 2.1 1.8
4 3.2 2.8 2.4
5 4.0 3.5 3.0
6 4.8 4.2 3.6
27
727
734
741
748
754
761
768
775
782
789
7 5.6 4.9 4.2
28
796
803
810
817
824
831
837
844
851
858
8 6.4 5.6 4.8
29
865
872
879
886
893
900
906
913
920
927
0 7.2 6.3 5.4
630
934
941
948
955
962
969
975
982
989
996
31
80003
010
017
024
030
037
044
051
058
065
32
072
079
085
092
099
106
113
120
127
134
33
140
147
154
161
168
175
182
188
195
202
34
209
216
223
229
236
243
250
257
264
271
35
277
284
291
298
305
312
318
325
332
339
36
346
353
359
366
373
380
387
393
400
407
37
414
421
428
434
441
448
455
462
468
475
38
482
489
496
502
509
516
523
530
536
543
39
550
557
564
570
577
584
591
598
604
611
640
618
625
632
638
645
652
659
665
672
679
_
41
686
693
699
706
713
720
726
733
740
747
42
754
760
767
774
781
787
794
801
808
814
43
821
828
835
841
848
855
862
868
875
882
44
889
895
902
909
916
922
929
936
943
949
45
956
963
969
976
983
990
996
*003
*010
*017
46
81023
030
037
043
050
057
064
070
077
084
47
090
097
104
111
117
124
131
137
144
151
48
158
164
171
178
184
191
198
204
211
218
49
224
231
238
245
251
258
265
271
278
285
650
291
298
305
311
318
325
331
338
345
a5i
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
189
0
1
2
3
4
5 | 6
7
8
9
Prop. Pts.
650
81291
298
305
311
318
325
331
338
345
351
51
358
365
371
378
385
391
398
405
411
418
52
425
431
438
445
451
458
465
471
478
485
53
491
498
505
511
518
525
531
538
544
551
54
558
564
571
578
584
591
598
604
611
617
55
624
631
637
644
651
657
664
671
677
684
56
690
697
704
710
717
723
730
737
743
750
57
757
763
770
776
783
790
796
803
809
816
58
823
829
836
842
849
856
862
869
875
882
59
889
895
902
908
915
921
928
935
941
948
660
954
961
968
974
981
987
994
*000
*007
*014
61
82020
027
033
040
046
053
060
066
073
079
62
086
092
099
105
112
119
125
132
138
145
63
151
158
164
171
178
184
191
197
204
210
64
217
223
230
236
243
249
256
263
269
276
65
282
289
295
302
308
315
321
328
334
341
66
347
354
360
367
373
380
387
393
400
406
67
413
419
426
432
439
445
452
458
465
471
68
478
484
491
497
504
510
517
523
530
536
69
543
549
556
562
569
575
582
5S8
595
601
670
607
614
620
627
633
640
646
653
659
666
71
672
679
685
692
698
705
711
718
724
730
7 6
72
737
743
750
756
763
769
776
782
789
795
1 0.7 0.6
73
802
808
814
821
827
834
840
847
853
860
2 1.4 1.2
74
866
872
879
885
892
898
905
911
918
924
3 2.1 1-8
75
930
937
943
950
956
963
969
975
982
988
4 2.8 2.4
76
995
*001
*008
*014
*020
*027
*033
*040
*046
*052
5 3.5 3.0
6 4.2 3.6
77
83059
065
072
078
085
091
097
104
110
117
7 4.9 4.2
78
123
129
136
142
149
155
161
168
174
181
8 5.6 4.8
79
187
193
200
206
213
219
225
232
238
245
9 6.3 5.4
680
251
257
264
270
276
283
289
296
302
308
81
315
321
327
334
340
347
353
359
366
372
82
378
385
391
398
404
410
417
423
429
436
83
442
448
455
461
467
474
480
487
493
499
84
506
512
518
525
531
537
544
550
556
563
85
569
575
582
588
594
601
607
613
620
626
86
632
639
645
651
658
664
670
677
683
689
87
696
702
708
715
721
727
734
740
746
753
88
759
765
771
778
784
790
797
803
809
816
89
822
828
835
841
847
853
860
866
872
879
690
885
891
897
904
910
916
923
929
935
942
91
948
954
960
967
973
979
985
992
998
*004
92
84011
017
023
029
036
042
048
055
061
067
93
073
080
086
092
098
105
111
117
123
130
94
136
142
148
155
161
167
173
180
186
192
95
198
205
211
217
223
230
236
242
248
255
96
261
267
273
280
286
292
298
305
311
317
97
323
330
336
342
348
354
361
367
373
379
98
386
392
398
404
410
417
423
429
435
442
99
448
454
460
4(56
473
479
485
491
497
504
700
510
516
522
528
535
541
547
553
559
566
0
1
2
3
4
5
6
7 8
9
Prop, Pts.
190
Table 3. Number Logarithms
0
' 1
2
3
4
5
6
7
8
9
Prop. Pts.
700
84510
516
522
528
535
541
547
553
559
566
01
572
578
584
590
597
603
609
615
621
628
02
634
640
646
652
658
665
671
677
683
689
03
696
702
708
714
720
726
733
739
745
751
04
757
763
770
776
782
788
794
800
807
813
05
819
825
831
837
844
850
856
862
868
874
06
880
887
893
899
905
911
917
924
930
936
07
942
948
954
960
967
973
979
985
991
997
08
85003
009
016
022
028
034
040
046
052
058
09
065
071
077
083
089
095
101
107
114
120
710
126
132
138
144
150
156
163
169
175
181
11
187
193
199
205
211
217
224
230
236
242
12
248
254
260
266
272
278
285
291
297
303
13
309
315
321
327
333
339
345
352
358
364
14
370
376
382
388
394
400
406
412
418
425
15
431
437
443
449
455
461
467
473
479
485
16
491
497
503
509
516
522
528
534
540
546
17
552
558
564
570
576
582
588
594
600
606
18
612
618
625
631
637
643
649
655
661
667
19
673
679
685
691
697
703
709
715
721
727
720
733
739
745
751
757
763
769
775
781
788
21
794
800
806
812
818
824
830
836
842
848
765
22
854
860
866
872
878
884
890
896
902
908
1 0.7 0.6 0.5
23
914
920
926
932
938
944
950
956
962
968
2 1.4 1.2 1.0
24
25
26
974
86034
094
980
040
100
986
046
106
992
052
112
998
058
118
*OQ4
064
124
*010
070
130
*016
076
136
*022
082
141
*028
088
147
3 2.1 1.8 1.5
4 2.8 2.4 2.0
5 3.5 3.0 2.5
6 4.2 3.6 3.0
27
153
159
165
171
177
183
189
195
201
207
7 4.9 4.2 3.5
28
213
219
225
231
237
243
249
255
261
267
8 5.6 4.8 4.0
29
273
279
285
291
297
303
308
314
320
326
9 6.3 5.4 4.5
730
332
338
344
350
356
362
368
374
380
386
31
392
398
404
410
415
421
427
433
439
445
32
451
457
463
469
475
481
487
493
499
504
33
510
516
522
528
534
540
546
552
558
564
34
570
576
581
587
593
599
605
611
617
623
35
629
635
641
646
652
658
664
670
676
682
36
688
694
700
705
711
717
723
729
735
741
37
747
753
759
764
770
776
782
788
794
800
38
806
812
817
823
829
835
841
847
853
859
39
864
870
876
882
888
894
900
906
911
917
740
923
929
935
941
947
953
958
964
970
976
41
982
988
994
999
*005
*011
*017
*023
*029
*035
42
87040
046
052
058
064
070
075
081
087
093
43
099
105
111
116
122
128
134
140
146
151
44
157
163
169
175
181
186
192
198
204
210
45
216
221
227
233
239
245
251
256
262
268
46
274
280
286
291
297
303
309
315
320
326
47
332
338
344
349
355
361
367
373
379
384
48
390
396
402
408
413
419
425
431
437
442
49
448
454
460
466
471
477
483
489
495
500
750
506
512
518
523
529
535
541
547
552
558
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
191
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
750
87506
512
518
523
529
535
541
547
552
558
51
564
570
576
581
587
593
599
604
610
616
52
622
628
633
639
645
651
656
662
668
674
53
679
685
691
697
703
708
714
720
726
731
54
737
743
749
754
760
766
772
777
783
789
55
795
800
806
812
818
823
829
835
841
846
56
852
858
864
869
875
881
887
892
898
904
57
910
915
921
927
933
938
944
950
955
961
58
967
973
978
984
990
996
*001
*007
*013
*018
59
88024
030
036
041
047
053
058
064
070
076
760
081
087
093
098
104
110
116
121
127
133
61
138
144
150
156
161
167
173
178
184
190
62
195
201
207
213
218
224
230
235
241
247
63
252
258
264
270
275
281
287
292
298
304
64
309
315
321
326
332
338
343
349
355
360
65
366
372
377
383
389
395
400
406
412
417
66
423
429
434
440
446
451
457
463
468
474
67
480
485
491
497
502
508
513
519
525
530
68
536
542
547
553
559
564
570
576
581
587
69
593
598
604
610
615
621
627
632
638
643
770
649
655
660
666
672
677
683
689
694
700
71
705
711
717
722
728
734
739
745
750
756
6 5
72
762
767
773
779
784
790
795
801
807
812
1 0.6 0.5
73
818
824
829
835
840
846
852
857
863
868
2 1.2 1.0
74
75
76
874
930
986
880
936
992
885
941
997
891
947
*003
897
953
*009
902
958
*014
908
964
*020
913
969
*025
919
975
*031
925
981
*037
3 1.8 1.5
4 2.4 2.0
5 3.0 2.5
6 3.6 3.0
77
89042
048
053
059
064
070
076
081
087
092
7 4.2 3.5
78
098
104
109
115
120
126
131
137
143
148
8 4.8 4.0
79
154
159
165
170
176
182
187
193
198
204
9 5.4 4.5
780
209
215
221
226
232
237
243
248
254
260
81
265
271
276
282
287
293
298
304
310
315
82
321
326
332
337
343
348
354
360
365
371
83
376
382
387
393
398
404
409
415
421
426
84
432
437
443
448
454
459
465
470
476
481
85
487
492
498
504
509
515
520
526
531
537
86
542
548
553
559
564
570
575
581
586
592
1 9
87
597
603
609
614
620
625
631
636
642
647
88
653
658
664
669
675
680
686
691
697
702
89
708
713
719
724
730
735
741
'746
752
757
790
763
768
774
779
785
790
796
801
807
812
91
818
823
829
834
840
845
851
856
862
867
92
873
878
883
889
894
900
905
911
916
922
93
927
933
938
944
949
955
960
966
971
977
94
982
988
993
998
*004
*009
*015
*020
*026
*031
95
90037
042
048
053
059
064
069
075
080
086
96
091
097
102
108
113
119
124
129
135
140
97
146
151
157
162
168
173
179
184
189
195
98
200
206
211
217
222
227
233
238
244
249
99
255
260
266
271
276
282
287
293
298
304
800
309
314
320
325
331
336
342
347
352
358
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
192
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
800
90309
314
320
325
331
336
342
347
352
358
01
363
369
374
380
385
390
396
401
407
412
02
417
423
428
434
439
445
450
455
461
466
03
472
477
482
488
493
499
504
509
515
520
04
526
531
536
542
547
553
558
563
569
574
05
580
585
590
596
601
607
612
617
623
628
06
634
639
644
650
655
660
666
671
677
682
07
687
693
698
703
709
714
720
725
730
736
08
741
747
752
757
763
768
773
779
784
789
09
795
800
806
811
816
822
827
832
838
843
810
849
854
859
865
870
875
881
886
891
897
11
902
907
913
918
924
929
934
940
945
950
12
956
961
966
972
977
982
988
993
998
*004
13
91009
014
020
025
030
036
041
046
052
057
14
062
068
073
078
084
089
094
100
105
110
15
116
121
126
132
137
142
148
153
158
164
16
169
174
180
185
190
196
201
206
212
217
17
222
228
233
238
243
249
254
259
265
270
18
275
281
286
291
297
302
307
312
318
323
19
328
334
339
344
350
355
360
365
371
376
820
381
387
392
397
403
408
413
418
424
429
21
434
440
445
450
455
461
466
471
477
482
6 5
22
487
492
498
503
508
514
519
524
529
535
1 0.6 0.5
23
•540
545
551
556
561
566
572
577
582
587
2 1.2 1.0
24
25
593
645
598
651
603
656
609
601
614
666
619
672
624
677
630
682
635
687
640
693
3 1.8 1.5
4 2.4 2.0
26
698
703
709
714
719
724
730
735
740
745
5 3.0 2.5
6 3.6 3.0
27
751
756
761
766
772
777
782
787
793
798
7 4.2 3.5
28
803
808
814
819
824
829
834
840
845
850
8 4.8 4.0
29
855
861
866
871
876
882
887
892
897
903
9 5.4 4.5
830
908
913
918
924
929
934
939
944
950
<t55
31
960
965
971
976
981
986
991
997
*002
*007
32
92012
018
023
028
033
038
044
049
054
059
33
065
070
075
080
085
091
096
101
106
111
34
117
122
127
132
137
143
148
153
158
163
35
169
174
179
184
189
195
200
205
210
215
36
221
226
231
236
241
247
252
257
262
267
37
273
278
283
288
293
298
304
309
314
319
38
324
330
335
340
345
350
355
361
366
371
39
376
381
387
392
3!)7
402
407
412
418
423
840
428
433
438
443
449
454
459
464
469
474
•
41
480
485
490
495
500
505
511
516
521
526
42
531
536
542
547
552
557
562
567
572
578
43
583
588
593
598
603
609
614
619
624
629
44
634
639
645
650
655
660
665
670
675
681
45
686
691
696
701
706
711
716
722
727
732
46
737
742
747
752
758
763
768
773
778
783
47
788
793
799
804
809
814
819
824
829
&34
48
840
845
850
855
860
865
870
875
881
886
49
891
896
901
906
911
916
921
927
932
937
850
942
947
952
957
%2
967
973
978
983
988
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
193
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
850
92942
947
952
957
962
967
973
978
983
988
51
993
998
«003
*008
*013
*018
«024
«029
*034
*039
52
93044
049
054
059
064
069
075
080
085
090
53
095
100
105
110
115
120
125
131
136
141
54
146
151
156
161
166
171
176
181
186
192
55
197
202
207
212
217
222
227
232
237
242
56
247
252
258
263
268
273
278
283
288
293
57
298
303
308
313
318
323
328
334
339
344
58
349
354
359
364
369
374
379
384
389
394
.
59
399
404
409
414
420
425
430
435
440
445
860
450
455
460
465
470
475
480
485
490
495
61
500
505
510
515
520
526
531
536
541
546
62
551
556
561
566
571
576
581
586
591
596
63
601
606
611
616
621
626
631
636
641
646
64
651
656
661
666
671
676
682
687
692
697
65
702
707
712
717
722
727
732
737
742
747
66
752
757
762
767
772
777
782
787
792
797
67
802
807
812
817
822
827
832
837
842
847
68
852
857
862
867
872
877
882
887
892
897
69
902
907
912
917
922
927
932
937
942
947
870
952
957
962
967
972
977
982
987
992
997
71
94002
007
012
017
022
027
032
037
042
047
654
72
052
057
062
067
072
077
082
086
091
096
1 0.6 0.5 0.4
73
101
106
111
116
121
126
131
136
141
146
2 1.2 1.0 0.8
74
75
76
151
201
250
156
206
255
161
211
260
166
216
265
171
221
270
176
226
275
181
231
280
186
236
285
191
240
290
196
245
295
3 1.8 1.5 1.2
4 2.4 2.0 1.6
5 3.0 2.5 2.0
6 3.6 3.0 2.4
77
300
305
310
315
320
325
330
335
340
345
7 4.2 3.5 2.8
78
349
354
359
364
369
374
379
384
389
394
8 4.8 4.0 3.2
79
399
404
409
414
419
424
429
433
438
443
9 5.4 4.5 3.6
880
448
453
458
463
4(58
473
478
483
488
493
81
498
503
507
512
517
522
527
532
537
542
82
547
552
557
5(>2
567
571
576
581
58(5
591
83
596
601
606
611
616
621
626
630
635
640
84
645
650
655
660
665
670
675
680
685
689
85
694
699
704
709
714
719
724
729
734
738
86
743
748
753
758
763
768
773
778
783
787
87
792
797
802
807
812
817
822
827
832
836
88
841
846
851
856
861
866
871
876
880
885
89
890
895
900
905
910
915
919
924
929
934
890
939
944
949
954
959
963
968
973
978
983
91
988
993
998
«002
*007
*012
*017
*022
«027
*032
92
95036
041
046
051
056
061
066
071
075
080
93
085
090
095
100
105
109
114
119
124
129
94
134
139
143
148
153
158
163
168
173
177
95
182
187
192
197
202
207
211
216
221
226
96
231
236
240
245
250
255
260
265
270
274
97
279
284
289
294
299
303
308
313
318
323
98
328
332
337
342
347
352
357
361
366
371
99
376
381
386
390
395
400
405
410
415
419
900
424
429,
434
439
444
448
453
458
463
468
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
194
Table 3. Number Logarithms
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
900
95424
429
434
439
444
448
453
458
463
468
01
472
477
482
487
492
497
501
506
511
516
02
521
525
530
535
540
545
550
554
559
564
03
569
574
578
583
588
593
598
602
607
612
04
617
622
626
631
636
641
646
650
655
660
05
665
670
674
679
684
689
694
698
703
708
06
713
718
722
727
732
737
742
746
751
756
07
761
766
770
775
780
785
789
794
799
804
08
809
813
818
823
828
832
837
842
847
852
09
85H
861
866
871
875
880
885
890
895
899
910
904
909
914
918
923
928
933
938
942
947
11
952
957
961
966
971
976
980
985
990
995
12
999
*004
*009
*014
*019
*023
*028
*033
*038
*042
13
96.047
052
057
061
066
071
076
080
085
090
14
095
099
104
109
114
118
123
128
133
137
15
142
147
152
156
161
166
171
175
180
185
16
190
194
199
204
209
213
218
223
227
232
17
237
242
246
251
256
261
265
270
275
280
18
284
289
294
298
303
308
313
317
322
327
19
332
336
341
346
350
355
360
365
369
374
920
379
384
388
393
398
402
407
412
417
421
21
426
431
435
440
445
450
454
459
464
468
5 4
22
473
478
483
487
492
497
501
506
511
515
1 0.5 0.4
23
520
525
530
534
539
544
548
553
558
562
2. 1.0 0.8
24
567
572
577
581
586
591
595
600
605
609
3 1.5 1.2
49 fi 1 fi
25
614
619
624
628
633
638
642
647
652
656
~.\i i ')
50 e OH
26
661
666
670
675
680
685
689
694
699
703
_.-> —•'/
6 3.0 2.4
27
708
713
717
722
727
731
736
741
745
750
7 3.5 2.8
28
755
759
764
769
774
778
783
788
792
797
8 4.0 3.2
29
802
806
811
816
820
825
830
834
839
844
9 4.5 3.6
930
848
853
858
862
867
872
876
881
886
890
31
895
900
904
909
914
918
923
928
932
937
32
942
946
951
956
960
965
970
974
979
984
33
988
993
997
*002
*007
*011
*016
*021
«025
*030
34
97035
039
044
049
053
058
063
067
072
077
35
081
086
090
095
100
104
109
114
118
123
36
128
132
137
142
146
151
155
160
165
169
37
174
179
183
188
192
197
202
206
211
216
38
220
225
230
234
239
243
248
253
257
262
39
267
271
276
280
285
290
294
299
304
308
940
313
317
322
327
331
336
340
345
350
354
41
359
364
368
373
377
382
387
391
396
400
42
405
410
414
419
424
428
433
437
442
447
43
451
456
460
465
470
474
479
483
488
493
44
497
502
506
511
516
520
525
529
534
539
45
543
548
552
557
562
566
571
575
580
585
46
589
594
598
603
607
612
617
621
626
630
47
635
640
644
649
653
658
663
667
672
676
48
681
685
690
695
699
704
708
713
717
722
49
727
731
736
740
745
749
754
759
763
768
950
772
777
782
786
791
795
800
804
809
813
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
Table 3. Number Logarithms
195
0
1
2
3
4
5
6
7
8
9
Prop. Pts.
950
97772
777
782
786
791
7ft5
800
804
809
813
51
818
823
827
832
836
841
845
850
855
859
52
864
868
873
877
882
886
891
896
900
905
53
909
914
918
923
928
932
937
941
946
950
54
955
959
964
968
973
978
982
987
991
996
55
98000
005
009
014
019
023
028
032
037
041
56
046
050
055
059
064
068
073
078
082
087
57
091
096
100
105
109
114
118
123
127
132
58
137
141
146
150
155
159
164
168
173
177
59
182
186
191
195
200
204
209
214
218
223
960
227
232
236
241
245.
250
254
259
263
268
61
272
277
281
286
290
295
299
304
308
313
62
318
322
327
331
336
340
345
349
354
358
63
363
367
372
376
381
385
390
394
399
403
64
408
412
417
421
426
430
435
439
444
448
65
453
457
462
466
471
475
480
484
489
493
66
498
502
507
511
516
520
525
529
534
538
67
543
547
552
556
561
565
570
574
579
583
68
588
592
597
601
605
610
614
619
623
628
69
632
637
641
646
650
655
659
664
668
673
970
677
682
686
691
695
700
704
709
713
717
71
722
726
731
735
740
744
749
753
758
762
5 4
72
767
771
776
780
784
789
793
798
802
807
1 0.5 0.4
73
811
816
820
825
829
834
838
843
847
851
2 1.0 0.8
74
856
860
865
869
874
878
883
887
892
896
3 1.5 1.2
75
900
905
909
914
918
923
927
932
936
941
4 2.0 1.6
76
945
949
954
958
963
967
972
976
981
985
5 2.5 2.0
6 3.0 2.4
77
989
994
998
*003
«007
*012
«016
*021
*025
*029
7 3.5 2.8
78
99034
038
043
047
052
056
061
065
069
074
8 4.0 3.2
79
078
083
087
092
096
100
105
109
114
118
9 4.5 3.6
980
123
127
131
136
140
145
149
154
158
162
81
167
171
176
180
185
189
193
198
202
207
82
211
216
220
224
229
233
238
242
247
251
83
255
260
264
269
273
277
282
286
291
295
84
300
304
308
313
317
322
326
330
335
339
85
344
348
352
357
361
366
370
374
379
383
86
388
392
396
401
405
410
414
419
423
427
87
432
436
441
445
449
454
458
463
467
471
88
476
480
484
489
493
498
502
506
511
515
89
520
524
528
533
537
542
546
550
555
559
990
564
568
572
577
581
585
590
594
599
603
91
607
612
616
621
625
629
634
638
642
647
92
651
656
660
664
669
673
677
682
686
691
93
695
699
704
708
712
717
721
726
730
734
94
739
743
747
752
756
760
765
769
774
778
95
782
787
791
795
800
804
808
813
817
822
96
826
830
835
839
843
848
852
856
861
865
97
870
874
878
883
887
891
896
900
904
909
98
913
917
922
926
930
935
939
944
•948
952
99
957
961
965
970
974
978
983
987
991
996
1000
000(10
004
009
013
017
022
026
030
035
039
0
1
2
3
4
5
6
7
8
9
Prop. Pta.
196
Table 4. Trigonometric Logarithms
0° (180°)
(359°) 179°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
—
0.00 000
—
—
0.00 000
—
60
1
6.46 373
.00 000
6.46 373
3.53 627
.00 000
3.53 627
59
2
6.76 476
.00 000
6.76 476
3.23 524
.00 000
.23 524
58
3
6.94 085
.00 000
6.94 085
3.05 915
.00 000
.05 915
57
4
7.06 579
.00 000
7.06 579
2.93 421
.00 000
2.93 421
56
5
7.16270
0.00 000
7.16270
2.83 730
0.00 000
2.83 730
55
6
.24 188
.00 000
.24 188
.75 812
.00 000
.75 812
54
7
.30 882
.00 000
.30 882
.69 118
.00000
.69 118
53
8
.36 682
.00 000
.36 682
.63 318
.00 000
.63 318
52
9
.41 797
.00 000
.41 797
.58 203
.00 000
.58 203
51
10
7.46 373
0.00 000
7.46 373
2.53 627
0.00 000
2.53 627
50
11
.50 512
.00 000
.50 512
.49 488
.00 000
.49 488
49
12
.54 291
.00 000
.54 291
.45 709
.00 000
.45 709
48
13
.57 767
.00 000
.57 767
.42 233
.00 000
.42 233
47
14
.60 985
.00 000
.60 986
• .39 014
.00 000
.39 015
46
15
7.63 982
0.00 000
7.63 982
2.36018
0.00 000
2.36018
45
16
.66 784
.00 000
.66 785
.33 215
.00 000
.33 216
44
17
.69417
9.99 999
.69 418
.30 582
.00 001
.30 583
43
18
.71 900
.99 999
.71 900
.28 100
.00 001
.28 100
42
19
.74 248
.99 999
.74 248
.25 752
.00 001
.25 752
41
20
7.76 475
9.99 999
7.76 476
2.23 524
0.00 001
2.23 525
40
21
.78 594
.99 999
.78 595
.21 405
.00 001
.21 406
39
22
.80 615
.99 999
.80 615
.19 385
.00 001
.19 385
38
23
.82 545
.99 999
.82 546
.17 454
.00 001
.17 455
37
24
.84393
.99 999
.84 394
.15 606
.00 001
.15 607
36
25
7.86 166
9.99 999
7.86 167
2.13 833
0.00 001
2.13 834
35
26
.87 870
.99 999
.87 871
.12 129
.00 001
.12 130
34
27
.89 509
.99 999
.89 510
.10 490
.00 001
.10491
33
28
.91 088
.99 999
.91 089
.08911
.00 001
.08 912
32
29
.92 612
.99 998
.92 613
.07 387
.00 002
.07 388
31
30
7.94 084
9.99 998
7.94 086
2.05 914
0.00 002
2.05 916
30
31
.95 508
.99 998
.95 510
.04 490
.00 002
.04 492
29
32
.96 887
.99 998
.96 889
.03 111
.00 002
.03 113
28
33
.98 223
.99 998
.98 225
.01 775
.00 002
.01 777
27
34
.99 520
.99 998
.99 522
.00 478
.00 002
.00 480
26
35
8.00 779
9.99 998
8.00 781
1.99219
0.00 002
1.99 221
25
36
.02 002
.99 998
.02 004
.97 996
.00 002
.97 998
24
37
.03 192
.99 997
.03 194
.96 806
.00 003
.96 808
23
38
.04350
.99 997
.04353
.95 647
.00 003
.95 650
22
39
.05 478
.99 997
.05 481
.94 519
.00 003
.94 522
21
40
8.06 578
9.99 997
8.06 581
1.93419
0.00 003
1.93 422
20
41
.07 650
.99 997
.07 653
.92 347
.00 003
.92 350
19
42
.08 696
.99 997
.08 700
.91 300
.00 003
.91 304
18
43
.09 718
.99 997
.09 722
.90 278
.00 003
.90 282
17
44
.10717
.99 996
.10 720
.89 280
.00 004
.89 283
16
45
8.11 693
9.99 996
8.11 696
1.88 304
0.00 004
1.88 307
15
46
.12 647
.99 996
.12 651
.87 349
.00 004
.87 353
14
47
.13 581
.99 996
.13 585
.86 415
.00004
.86 419
13
48
.14495
.99 996
.14 500
.85 500
.00 004
.85 505
12
49
.15391
.99 996
.15 395
.84605
.00 004
.84609
11
50
8.16 268
9.99 995
8.16 273
1.83 727
0.00 005
1.83 732
10
51
.17 128
.99 995
.17 133
.82 867
.00 005
.82 872
9
52
.17971
.99 995
.17 976
.82 024
.00 005
.82 029
8
53
.18 798
.99 995
.18 804
.81 196
.00 005
.81 202
7
54
.19610
.99 995
.19616
.80 384
.00 005
.80 390
6
55
8.20 407
9.99 994
8.20413
1.79587
0.00 006
1.79 593
5
56
.21 189
.99 994
.21 195
.78 805
.00 006
.78811
4
57
.21 958
.99 994
.21 964
.78 036
.00 006
.78 042
3
58
.22 713
.99 994
.22 720
.77 280
.00 006
.77 287
2
59
.23 456
.99 994
.23 462
.76 538
.00 006
.76 544
1
60
8.24 186
9.99 993
8.24 192
1.75 808
0.00 007
1.75 814
0
Cos
Sin
Cot
Tan
Csc
Sec
'
90° (270°)
(269°) 89°
Table 4. Trigonometric Logarithms
197
1° (181°)
(358°) 178°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
8.24 186
9.99 993
8.24 192
1.75 808
0.00 007
1.75 814
60
1
.24 903
.99 993
.24 910
.75 090
.00 007
.75 097
59
2
.25 609
.99 993
.25 616
.74384
.00007
.74 391
58
3
.26 304
.99 993
.26 312
.73 688
.00 007
.73 696
57
4
.26 988
.99 992
.26 996
.73 004
.00008
.73 012
56
5
8.27 661
9.99 992
8.27 669
1.72 331
0.00 008
1.72 339
55
6
.28 324
.99 992
.28 332
.71 668
.00008
.71 676
54
7
.28 977
.99 992
.28 986
.71 014
.00008
.71 023
53
8
.29 621
.99 992
.29 629
.70 371
.00008
.70 379
52
9
.30 255
.99 991
.30 263
.69 737
.00009
.69 745
51
10
8.30 879
9.99 991
8.30 888
1.69 112
0.00 009
1.69 121
50
11
.31 495
.99 991
.31 505
.68 495
.00009
.68 505
49
12
.32 103
.99 990
.32 112
.67 888
.00010
.67 897
48
13
.32 702
.99 990
.32 711
.67 289
.00 010
.67 298
47
14
.33 292
.99 990
.33 302
.66 698
.00010
.66 708
46
15
8.33 875
9.99 990
8.33 886
1.66 114
0.00 010
1.66 125
45
16
.34 450
.99 989
.34 461
.65 539
00.011
65550
44
17
.35 018
.99 989
.35 029
.64971
.00011
.64982
43
18
.35 578
.99 989
.35 590
.64410
.00011
.64 422
42
'19
.36 131
.99 989
.36 143
.63 857
.00011
.63 869
41
20
8.36 678
9.99 988
8.36 689
1.63311
0.00 012
1.63 322
40
21
.37 217
.99 988
.37 229
.62 771
.00 012
.62 783
39
22
.37 750
.99 988
.37 762
.62 238
.00 012
.62 250
38
23
.38 276
.99 987
.38 289
.61 711
.00 013
.61 724
37
24
.38 796
.99 987
.38809
.61 191
.00 013
.61 204
36
25
8.39 310
9.99 987
8.39 323
1.60 677
0.00 013
1.60 690
35
26
.39 818
.99 986
.39 832
.60 168
.00014
.60 182
34
27
.40 320
.99 986
.40 334
.59 666
.00 014
.59 680
33
28
.40 816
.99 986
.40 830
.59 170
.00 014
.59 184
32
29
.41 307
.99 985
.41 321
.58 679
.00015
.58 693
31
30
8.41 792
9.99 985
8.41 807
1.58 193
0.00 015
1.58 208
30
31
.42 272
.99 985
.42287
.57 713
.00 015
.57 728
29
32
.42 746
.99984
.42 762
.57 238
.00 016
.57 254
28
33
.43 216
.99984
.43 232
.56 768
.00 016
.56 784
27
34
.43 680
.99984
.43 696
.56 304
.00016
.56 320
26
35
8.44 139
9.99 983
8.44 156
1.55 844
0.00 017
1.55861
25
36
.44594
.99983
.44611
.55 389
.00017
.55 406
24
37
.45044
.99 983
.45 061
.54 939
.00 017
.54 956
23
38
.45 489
.99 982
.45 507
.54493
.00018
.54511
22
39
.45 930
.99 982
.45 948
.54 052
.00018
.54 070
21
40
8.46 366
9.99 982
8.46 385
1.53 615
0.00 018
1.53 634
20
41
.46 799
.99 981
.46 817
.53 183
.00 019
.53 201
19
42
.47 226
.99 981
.47 245
.52 755
.00019
.52 774
18
43
.47 650
.99 981
.47 669
.52 331
.00019
.52 350
17
44
.48 069
.99 980
.48 089
.51911
.00020
.51 931
16
45
8.48 485
9.99 980
8.48 505
1.51 495
0.00 020
1.51 515
15
46
.48 896
.99 979
.48 917
.51 083
.00 021
.51 104
14
47
.49 304
.99 979
.49325
.50 675
.00021
.50 696
13
48
.49 708
.99 979
.49 729
.50 271
.00 021
.50 292
12
49
.50 108
.99 978
.50 130
.49 870
.00022
.49 892
11
50
8.50 504
9.99 978
8.50 527
1.49 473
0.00 022
1.49 496
10
51
.50 897
.99 977
.50 920
.49 080
.00023
.49 103
9
52
.51 287
.99 977
.51 310
.48 690
.00023
.48713
8
53
.51 673
.99 977
.51 696
.48 304
.00 023
.48 327
7
54
.52 055
.99 976
.52 079
.47 921
.00024
.47 945
6
55
8.52 434
9.99 976
8.52 459
1.47 541
0.00 024
1.47 566
5
56
.52 810
.99 975
.52 835
.47 165
.00025
.47 190
4
57
.53183
.99 975
.53 208
.46 792
.00025
.46 817
3
58
.53 552
.99 974
.53 578
.46 422
.00026
.46448
2
59
.53 919
.99 974
.53 945
.46 055
.00026
.46 081
1
60
8.54 282
9.99 974
8.54 308
1.45692
0.00 026
1.45 718
0
Cos
Sin
Cot
Tan
Csc
Sec
'
91° (271°)
(268°) 88°
198
Table 4. Trigonometric Logarithms
2° (182°)
(357°) 177°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
8.54 282
9.99 974
8.54 308
1.45692
0.00 026
1.45718
60
1
.54 642
.99 973
.54 669
.45 331
.00 027
.45 358
59
2
.54 999
.99 973
.55 027
.44 973
.00 027
.45 001
58
3
.55 354
.99 972
.55 382
.44 618
.00 028
.44 646
57
4
.55 705
.99 972
.55 734
.44 266
.00 028
.44 295
56
5
8.56 054
9.99 971
8.56 083
1.43917
0.00 029
1.43 946
55
6
.56 400
.99 971
.56 429
.43 571
.00 029
.43 600
54
7
.56 743
.99 970
.56 773
.43 227
.00030
.43 257
53
8
.57 084
.99 970
.57 114
.42 886
.00 030
.42 916
52
9
.57 421
.99 969
.57 452
.42 548
.00 031
.42 579
51
10
8.57 757
9.99 969
8.57 788
1.42 212
0.00 031
1.42 243
50
11
.58 089
.99968
.58 121
.41 879
.00 032
.41 911
49
12
.58 419
.99 968
.58 451
.41 549
.00 032
.41 581
48
13
.58 747
.99 967
.58 779
.41 221
.00 033
.41 253
47
14
.59 072
.99 967
.59 105
.40 895
.00 033
.40 928
46
15
8.59 395
9.99 967
8.59 428
1.40 572
0.00 033
1.40 605
45
16
.59 715
.99 966
.59 749
.40 251
.00 034
.40 285
44
17
.60 033
.99 966
.60 068
.39 932
.00 034
.39 967
43
18
.60 349
.99 965
.60384
.39616
.00 035
.39 651
42
19
.60 662
.99 964
.60 698
.39 302
.00 036
.39 338
41
20
8.60 973
9.99 964
8.61 009
1.38991
0.00 036
1.39 027
40
21
.61 282
.99 963
.61 319
.38 681
.00 037
.38 718
39
22
.61 589
.99 963
.61 626
.38 374
.00 037
.38411
38
23
.61 894
.99 962
.61 931
.38 069
.00 038
.38 106
37
24
.62 196
.99 962
.62 234
.37 766
.00 038
.37 804
36
25
8.62 497
9.99 961
8.62 535
1.37465
0.00 039
1.37503
35
26
.62 795
.99 961
.62 834
.37 166
.00 039
.37 205
34
27
.63 091
.99 960
.63 131
.36 869
.00 940
.36 909
33
28
.63 385
.99 960
.63 426
.36 574
.00 040
.36 615
32
29
.63 678
.99 959
.63 718
.36 282
.00 041
.36 322
31
30
8.63 968
9.99 959
8.64 009
1.35 991
0.00 041
1.36 032
30
31
.64256
.99 958
.64298
.35 702
.00 042
.35 744
29
32
.64 543
.99 958
.64585
.35 415
.00 042
.35 457
28
33
.64827
.99 957
.64870
.35 130
.00043
.35 173
27
34
.65 110
.99 956
.65 154
.34846
.00044
.34 890
26
35
8.65 391
9.99 956
8.65 435
1.34 565-
0.00 044
1.34 609
25
36
.65 670
.99 955
.65 715
.34285
.00 045
.34 330
24
37
.65 947
.99 955
.65 993
.34 007
.00045
.34 053
23
38
.66 223
.99 954
.66 269
.33 731
.00 046
.33 777
22
39
.66 497
.99 954
.66543
.33 457
.00 046
.33 503
21
40
8.66 769
9.99 953
8.66816
1.33 184
0.00 047
1.33 231
20
41
.67 039
.99 952
.67 087
.32 913
.00 048
.32 961
19
42
.67 308
.99 952
.67 356
.32 644
.00 048
.32 692
18
43
.67 575
.99 951
.67 624
.32 376
.00 049
.32 425
17
44
.67 841
.99 951
.67 890
.32 110
.00 049
.32 159
16
45
8.68 104
9.99 950
8.68 154
1.31 846
0.00 050
1.31 896
15
46
.68 367
.99 949
.68 417
.31 583
.00 051
.31 633
14
47
.68 627
.99 949
.68 678
.31 322
.00 051
.31 373
13
48
.68 886
.99 948
.68 938
.31 062
.00 052
.31 114
12
49
.69 144
.99 948
.69 196
.30 804
.00 052
.30 856
11
50
8.69 400
9.99 947
8.69 453
1.30 547
0.00 053
1.30 600
10
51
.69 654
.99 946
, .69 708
.30 292
.00 054
.30 346
9
52
.69 907
.99 946
.69 962
.30 038
.00 054
.30 093
8
53
.70 159
.99 945
.70 214
.29 786
.00 055
.29841
7
54
.70 409
.99944
.70 465
.29 535
.00 056
.29 591
6
55
8.70 658
9.99 944
8.70 714
1.29286
0.00 056
1.29 342
5
56
.70 905
.99 943
.70 962
.29 038
.00 057
.29 095
4
57
.71 151
.99 942
.71 208
.28 792
.00 058
.28849
3
58
.71 395
.99 942
.71 453
.28 547
.00 058
.28 605
2
59
.71 638
- .99 941
.71 697
.28 303
.00 059
.28 362
1
60
8.71 880
9.99 940
8.71 940
1.28060
0.00 060
1.28 120
0
Cos
Sin
Cot
1 Tan
Csc
Sec
'
92° (272°)
(267°) 87°
Table 4. Trigonometric Logarithms
199
3° (183°)
(356°) 176°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
8.71 880
9.99 940
8.71 940
1.28060
0.00 060
1.28 120
60
1
.72 120
.99 940
.72 181
.27 819
.00 060
.27 880
59
2
.72 359
.99 939
.72 420
.27 580
.00 061
.27641
58
3
.72 597
.99 938
.72 659
.27 341
.00062
.27 403
57
4
.72 834
.99 938
.72 896
. .27 104
.00062
.27 166
56
5
8.73 069
9.99 937
8.73 132
1.26 868
0.00 063
1.26 931
55
6
.73 303
.99 936
.73 366
.26 634
.00 064
.26 697
54
7
.73 535
.99 936
.73 600
.26400
.00 064
.26 465
53
8
.73 767
.99 935
.73832
.26 168
.00065
.26 233
52
9
.73 997
.99 934
.74 063
.25 937
.00 066
.26 003
51
10
8.74 226
9.99 934
8.74 292
1.25 708
0.00 066
1.25 774
50
11
.74 454
.99 933
.74 521
.25 479
.00067
.25 546
49
12
.74 680
.99 932
.74 748
.25 252
.00 068
.25 320
48
13
.74 906
.99 932
.74 974
.25 026
.00068
.25 094
47
14
.75 130
.99 931
.75 199
.24 801
.00069
.24 870
46
15
8.75 353
9.99 930
8.75 423
1.24 577
0.00 070
1.24 647
45
16
.75 575
.99 929
.75645
.24 355
.00071
.24 425
44
17
.75 795
.99 929
.75 867
.24 133
.00071
.24 205
43
18
.76 015
.99 928
.76 087
.23 913
.00072
.23 985
42
19
.76 234
.99 927
.76 306
.23 694
.00073
.23 766
41
20
8.76 451
9.99 926
8.76 525
1.23475
0.00 074
1.23 549
40
21
.76 667
.99 926
.76 742
.23 258
.00074
.23 333
39
22
.76 883
.99 925
.76 958
.23 042
.00 075
.23 117
38
23
.77 097
.99 924
.77 173
.22 827
.00076
.22 903
37
24
.77 310
.99 923
.77 387
.22 613
.00077
.22 690
36
25
8.77 522
9.99 923
8.77 600
1.22 400
0.00 077
1.22 478
35
26
.77 733
.99922
.77811
.22 189
.00078
.22 267
34
27
.77 943
.99 921
.78 022
.21 978
.00079
.22 057
33
28
.78 152
.99 920
.78 232
.21 768
.00080
.21848
32
29
.78 360
.99 920
.78441
.21 559
.00080
.21 640
31
30
8.78 568
9.99 919
8.78 649
1.21 351
0.00 081
1.21 432
30
31
.78 774
.99 918
.78 855
.21 145
.00082
.21 226
29
32
.78 979
.99 917
.79 061
.20 939
.00 083
.21 021
28
33
.79 183
.99 917
.79 266
.20 734
.00083
.20 817
27
34
.79 386
.99 916
.79 470
.20 530
.00084
.20 614
26
35
8.79 588
9.99 915
8.79 673
1.20327
0.00 085
1.20412
25
36
.79 789
.99 914
.79 875
.20 125
.00086
.20211
24
37
.79 990
.99 913
.80 076
.19 924
.00 087
.20 010
23
38
.80 189
.99 913
.80277
.19 723
.00 087
.19811
22
39
.80388
.99 912
.80476
.19524
• .00 088
.19612
21
40
8.80 585
9.99911
8.80 674
1.19326
0.00 089
1.19415
20
41
.80 782
.99 910
.80 872
.19 128
.00090
.19218
19
42
.80 978
.99 909
.81 068
.18 932
.00 091
.19 022
18
43
.81 173
.99 909
.81 264
.18 736
.00 091
.18 827
17
44
.81 367
.99 908
.81 459
.18541
.00092
.18 633
16
45
8.81 560
9.99 907
8.81 653
1.18 347
0.00 093
1.18440
15
46
.81 752
.99 906
.81 846
.18 154
.00 094
.18 248
14
47
.81 944
.99 905
.82 038
.17 962
.00095
.18 056
13
48
.82 134
.99904
.82 230
.17770
.00096
.17 866
12
49
.82 324
.99904
.82 420
.17 580
.00096
.17 676
11
50
8.82 513
9.99 903
8.82 610
1.17 390
0.00 097
1.17 487
10
51
.82 701
.99 902
.82 799
.17 201
.00 098
.17 299
9
52
.82888
.99 901
.82 987
.17013
.00 099
.17 112
8
53
.83 075
.99 900
.83175
.16 825
.00 100
.16 925
7
54
.83261
.99 899
.83361
.16 639
.00 101
.16 739
6
55
8.83 446
9.99 898
8.83 547
1.16453
0.00 102
1.16 554
5
•56
.83630
.99 898
.83732
.16 268
.00102
.16 370
4
57
.83 813
.99 897
.83916
.16084
.00103
.16 187
3
58
.83996
.99 896
.84100
.15 900
.00104
.16 004
2
59
.84177
.99 895
.84282
.15718
.00 105
.15 823
1
60
8.84 358
9.99 894
8.84 464
1.15536
0.00 106
1.15 642
0
Cos Sin
Cot
Tan
Csc
Sec
'
93° (273°)
(266°) 86°
200
Table 4. Trigonometric Logarithms
4° (184°)
(355°) 175°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
8.84 358
9.99 894
8.84464
1.15536
0.00 106
1.15642
60
1
.84539
.99 893
.84646
.15 354
.00 107
.15461
59
2
.84718
.99 892
.84826
.15 174
.00 108
.15 282
58 '
3
.84897
.99 891
.85006
.14 994
.00 109
.15 103
57
4
.85 075
.99 891
.85 185
.14 815
.00 109
.14 925
56
5
8.85 252
9.99 890
8.85 363
1.14 637
0.00 110
1.14748
55
6
.85 429
.99 889
.85 540
.14 460
.00 111
.14 571
54
7
.85 605
.99 888
.85717
.14 283
.00 112
.14 395
53
8
.85 780
.99 887
.85 893
.14 107
.00 113
.14 220
52
9
.85 955
.99 886
.86 069
.13 931
.00 114
.14 045
51
10
8.86 128
9.99 885
8.86 243
1.13 757
0.00 115
1.13 872
50
11
.86 301
.99884
.86 417
.13 583
.00 116
.13699
49
12
.86 474
.99 883
.86 591
.13 409
.00 117
.13526
48
13
.86 645
.99 882
.86 763
.13 237
.00 118
.13 355
47
14
.86 816
.99 881
.86 935
.13 065
.00 119
.13 184
46
15
8.86 987
9.99 880
8.87 106
1.12 894
0.00 120
1.13013
45
16
.87 156
.99 879
.87 277
.12 723
.00 121
.12844
44
17
.87 325
.99 879
.87 447
.12 553
.00 121
.12 675
43
18
.87 494
.99 878
.87 616
.12384
.00 122
.12 506
42
19
.87 661
.99 877
.87 785
.12215
.00 123
.12 339
41
20
8.87 829
9.99 876
8.87 953
1.12 047
0.00 124
1.12171
40
21
.87 995
.99 875
.88 120
.11 880
.00 125
.12 005
39
22
.88 161
.99 874
.88 287
.11 713
00126
.11 839
38
23
.88 326
.99 873
.88 453
.11 547
.00 127
.11 674
37
24
.88 490
.99 872
.88 618
.11 382
.00 128
.11 510
36
25
8.88 654
9.99 871
8.88 783
1.11217
0.00 129
1.11346
35
26
.88817
.99 870
.88 948
.11 052
.00 130
.11 183
34
27
.88 980
.99 869
.89 111
.10 889
.00 131
.11 020
33
28
.89 142
.99 868
.89 274
.10 726
.00 132
.10 858
32
29
.89 304
.99 867
.89 437
.10 563
.00 133
.10 696
31
30
8.89 464
9.99 866
8.89 598
1.10402
0.00 134
1.10 536
30
31
.89 625
.99 865
.89 760
.10 240
.00 135
.10375
29
32
.89784
.99 864
.89 920
.10 080
.00 136
.10216
28
33
.89 943
.99 863
.90 080
.09 920
.00 137
.10057
27
34
.90 102
.99 862
.90 240
.09760
.00 138
.09 898
26
35
8.90 260
9.99 861
8.90 399
1.09 601
0.00 139
1.09740
25
36
.90 417
.99 860
.90 557
.09 443
.00 140
.09 583
24
37
.90 574
.99 859
.90 715
.09 285
.00 141
.09 426
23
38
.90 730
.99 858
.90 872
.09 128
.00 142
.09 270
22
39
.90 885
.99 857
.91 029
.08 971
.00 143
.09 115
21
40
8.91 040
9.99 856
8.91 185
1.08 815
0.00 144
1.08 960
20
41
.91 195
.99 855
.91 340
.08 660
.00 145
.08 805
19
42
.91 349
.99 854
.91 495
.08 505
.00 146
.08 651
18
43
.91 502
.99 853
.91 650
.08 350
.00 147
.08 498
17
44
.91 655
.99 852
.91 803
.08 197
.00 148
.08 345
16
45
8.91 807
9.99 851
8.91 957
1.08043
0.00 149
1.08 193
15
46
.91 959
.99 850
.92 110
.07 890
.00 150
.08 041
14
47
.92 110
.99848
.92 262
.07 738
.00 152
.07 890
13
48
.92 261
.99 847
.92 414
.07 586
.00 153
.07 739
12
49
.92411
.99846
.92 565
.07 435
.00 154
.07 589
11
50
8.92 561
9.99 845
8.92 716
1.07 284
0.00 155
1.07439
10
51
.92 710
.99844
.92 866
.07 134
.00 156
.07 290
9
52
.92 859
.99 843
.93 016
.06984
.00 157
.07 141
8
53
.93 007
.99 842
.93 165
.06 835
.00 158
.06 993
7
54
.93 154
.99841
.93 313
.06 687
.00 159
.06846
6
55
8.93 301
9.99 840
8.93 462
1.06538
0.00 160
1.06 699
5
56
.93 448
.99 839
.93 609
.06 391
.00 161
.06 552
4
57
.93 594
.99 838
.93 756
.06 244
.00 162
.06 406
3
58
.93 740
.99 837
.93 903
.06 097
.00 163
.06 260
2
59
.93 885
.99 836
.94 049
.05 951
.00 164
.06 115
1
60
8,94 030
9.99 834
8.94 195
1.05 805
0.00 166
1.05970
0
Cos
Sin
Cot
Tan
Csc
Sec
'
94° (274°)
(265°) 85°
Table 4. Trigonometric Logarithms
201
5° (185°)
(354°) 174°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
8.94 030
9.99 834
8.94 195
1.05&05
0.00 166
1.05 970
60
1
.94174
.99 833
.94 340
.05 660
.00167
.05 826
59
2
.94 317
.99 832
.94485
.05 515
.00168
.05 683
58
3
.94461
.99831
.94 630
.05 370
.00 169
.05 539
57
4
.94603
.99830
.94 773
.05 227
.00170
.05 397
56
5
8.94 746
9.99 829
8.94 917
1.05 083
0.00 171
1.05 254
55
6
.94 887
.99 828
.95 060
.04940
.00172
.05 113
54
7
.95 029
.99 827
.95 202
.04798
.00173
.04971
53
8
.95 170
.99 825
.95344
.04656
.00 175
.04830
52
9
.95 310
.99 824
.95 486
.04514
.00 176
.04690
51
10
8.95 450
9.99 823
8.95 627
1.04 373
0.00 177
1.04 550
50
11
.95 589
.99 822
.95 767
.04233
.00178
.04411
49
12
.95 728
.99 821
.95 908
.04092
.00179
.04272
48
13
.95 867
.99 820
.96 047
.03 953
.00180
.04 133
47
14
.96005
.99 819
.96 187
.03 813
.00181
.03 995
46
15
8.96 143
9.99 817
8.96 325
1.03 675
0.00 183
1.03 857
45
16
.96 280
.99 816
.96464
.03 536
.00 184
.03 720
44
17
.96 417
.99 815
.96 602
.03 398
.00 185
.03 583
43
18
.96 553
.99 814
• .96 739
.03 261
.00 186
.03 447
42
19
.96 689
.99 813
.96 877
.03 123
.00187
.03 311
41
20
8.96 825
9.99 812
8.97 013
1.02987
0.00 188
1.03 175
40
21
.96 960
.99 810
.97 150
.02 850
.00190
.03040
39
22
.97 095
.99 809
.97285
.02 715
.00191
.02 905
38
23
.97 229
.99 808
.97 421
.02 579
.00 192
.02 771
37
24
.97 363
.99 807
.97 556
.02444
.00193
.02 637
36
25
8.97 496
9.99 806
8.97 691
1.02309
0.00 194
1.02 504
35
26
.97 629
.99804
.97 825
.02 175
.00196
.02 371
34
27
.97 762
.99 803
.97 959
.02041
.00197
.02 238
33
28
.97 894
.99 802
.98 092
.01 908
.00198
.02 106
32
29
.98 026
.99 801
.98 225
.01 775
.00199
.01 974
31
30
8.98 157
9.99 800
8.98 358
1.01 642
0.00 200
1.01 843
30
31
.98 288
.99 798
.98 490
.01 510
.00202
.01 712
29
32
.98 419
.99 797
.98 622
.01 378
.00 203
.01 581
28
33
.98549
.99 796
.98 753
.01 247
.00204
.01 451
27
34
.98 679
.99 795
.98884
.01 116
.00205
.01 321
26
35
8.98 808
9.99 793
8.99 015
1.00985
0.00 207
1.01 192
25
36
.98 937
.99 792
.99 145
.00 855
.00208
.01 063
24
37
.99066
.99 791
.99 275
.00725
.00209
.00934
23
38
.99 194
.99 790
.99 405
.00595
.00210
.00 806
22
39
.99 322
.99 788
.99 534
.00466
.00212
.00678
21
40
8.99 450
9.99 787
8.99 662
1.00338
0.00 213
1.00 550
20
41
.99 577
.99 786
.99 791
.00209
.00214
.00423
19
42
.99704
.99 785
.99 919
.00 081
.00 215
.00 296
18
43
.99830
.99783
9.00 046
0.99 954
.00 217
.00 170
17
44
.99 956
.99 782
.00 174
.99 826
.00218
.00044
16
45
9.00 082
9.99 781
9.00 301
0.99 699
0.00 219
0.99 918
15
46
.00207
.99780
.00 427
.99 573
.00 220
.99 793
14
47
.00332
.99 778
.00 553
.99 447
.00222
.99 668
13
48
.00456
.99 777
.00 679
.99 321
.00223
.99544
12
49
.00581
.99 776
.00 805
.99 195
.00224
.99 419
11
50
9.00704
9.99 775
9.00 930
0.99 070
0.00 225
0.99 296
10
51
.00828
.99 773
.01 055
.98 945
.00227
.99 172
9
52
.00951
.99 772
.01 179
.98 821
.00228
.99049
8
53
.01 074
.99 771
.01 303
.98 697
.00229
.98 926
7
54
.01 196
.99 769
.01 427
.98 573
.00231
.98804
6
55
9.01 318
9.99 768
9.01 550
0.98 450
0.00 232
0.98 682
5
56
.01440
.99 767
.01 673
.98 327
.00233
.98 560
4
57
.01 561
.99765
.01 796
.98204
.00 235
.98 439
3
58
.01 682
.99764
.01 918
.98 082
.00 236
.98318
2
59
.01803
.99 763
.02040
.97 960
.00237
.98 197
1
60
9.01 923
9.99 761
9.02 162
0.97 838
0.00 239
0.98 077
0
Cos
Sin
Cot
Tan
Csc
Sec
'
95° (275°)
(264°) 84°
202
Table 4. Trigonometric Logarithms
6° (186°)
(353°) 173C
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.01 923
9.99 761
9.02 162
0.97 838
0.00 239
0.98 077
60
1
.02 043
.99 760
.02 283
.97 717
00240
.97 957
59
2
.02 163
.99 759
.02 404
.97 596
.00 241
.97 837
58
3
.02 283
.99 757
.02 525
.97 475
.00 243
.97717
57
4
.02 402
.99 756
.02645
.97 355
.00 244
.97 598
56
5
9.02 520
9.99 755
9.02 766
0.97 234
0.00 245
0.97 480
55
6
.02 639
.99 753
.02 885
.97 115
.00 247
.97 361
54
7
.02 757
.99 752
.03 005
.96 995
.00 248
.97 243
53
8
.02 874
.99 751
.03 124
.96 876
.00 249
.97 126
52
9
.02 992
.99 749
.03 242
.96 758
.00 251
.97 008
51
10
9.03 109
9.99 748
9.03 361
0.96 639
0.00 252
0.96 891
50
11
.03 226
.99 747
.03 479
.96 521
.00 253
.96 774
49
12
.03 342
.99 745
.03 597
.96 403
.00 255
.96 658
48
13
.03 458
.99 744
.03 714
.96 286
.00 256
.96 542
47
14
.03 574
.99 742
.03 832
.96 168
.00 258
.96 426
46
15
9.03 690
9.99 741
9.03 948
0.96 052
0.00 259
0.96 310
45
16
.03 805
.99 740
.04 065
.95 935
.00 260
.96 195
44
17
.03 920
.99 738
.04 181
.95 819
.00 262
.96 080
43
18
.04034
.99 737
.04297
.95 703
.00 263
.95 966
42
19
.04 149
.99 736
.04413
.95 587
.00264
.95 851
41
20
9.04 262
9.99 734
9.04 528
0.95 472
0.00 266
0.95 738
40
21
.04376
.99 733
.04643
.95 357
.00 267
.95 624
39
22
.04 490
.99 731
.04758
.95 242
.00 269
.95 510
38
23
.04 603
.99 730
.04873
.95 127
.00 270
.95 397
37
24
.04 715
.99 728
.04987
.95 013
.00 272
.95 285
36
25
9.04 828
9.99 727
9.05 101
0.94 899
0.00 273
0.95 172
35
26
.04 940
.99 726
.05 214
.94 786
.00 274
.95 060
34
27
.05 052
.99 724
.05 328
.94 672
.00 276
.94 948
33
28
.05 164
.99 723
.05 441
. 94559
.00 277
.94 836
32
29
.05 275
.99721
.05 553
.94 447
.00 279
.94 725
31
30
9.05 386
9.99 720
9.05 666
0.94 334
0.00 280
0.94 614
30
31
.05 497
.99 718
.05 778
.94 222
.00 282
.94 503
29
32
.05 607
.99 717
.05 890
.94 110
.00 283
.94 393
28
33
.05 717
.99 716
.06 002
.93 998
.00284
.94 283
27 •
34
.05 827
.99 714
.06 113
.93 887
.00 286
.94 173
26
35
9.05 937
9.99 713
9.06 224
0.93 776
0.00 287
0.94 063
25
36
.06 046
.99711
.06 335
.93 665
.00 289
.93 954
24
37
.06 155
.99 710
.06 445
.93 555
.00 290
.93 845
23
38
.06264
.99 708
.06 556
.93 444
.00 292
.93 736
22
39
.06 372
.99 707
.06 666
.93 334
.00 293
.93 628
21
40
9.06 481
9.99 705
9.06 775
0.93 225
0.00 295
0.93 519
20
41
.06 589
.99 704
.06 885
.93 115
.00 296
.93411
19
42
.06 696
.99 702
.06 994
.93 006
.00 298
.93 304
18
43
.06 804
.99 701
.07 103
.92 897
.00 299
.93 196
17
44
.06911
.99 699
.07211
.92 789
.00 301
.93 089
16
45
9.07 018
9.99 698
9.07 320
0.92 680
0.00 302
0.92 982
15
46
.07 124
.99 696
.07 428
.92 572
.00304
.92 876
14
47
.07 231
.99 695
.07 536
.92 464
.00 305
.92 769
13
48
.07 337
.99 693
.07 643
.92 357
00.307
.92 663
12
49
.07 442
.99 692
.07 751
.92 249
.00 308
.92 558
11
50
9.07 548
9.99 690
9.07 858
0.92 142
0.00 310
0.92 452
10
51
.07 653
.99 689
..07 964
.92 036
.00311
.92 347
9
52
.07 758
.99 687
.08 071
.91 929
.00 313
.92 242
8
53
.07 863
.99 686
.08 177
.91 823
.00 314
.92 137
7
54
.07 968
.99 684
.08 283
.91 717
.00 316
.92 032
6
55
9.08 072
9.99 683
9.08 389
0.91 611
0.00 317
0.91 928
5
56
.08 176
.99 681
.08 495
.91 505
.00 319
.91 824
4
57
.08 280
.99 680
.08 600
.91 400
.00 320
.91 720
3
68
.08 383
.99 678
.08 705
.91 295
.00 322
.91 617
2
59
.08 486
.99 677
.08 810
.91 190
.00 323
.91 514
1
60
9.08 589
9.99 675
9.08 914
0.91 086
0.00 325
0.91 411
0
Cos
Sin
Cot
Tan
Csc
Sec
'
96° (276°)
(263°) 83°
Table 4. Trigonometric Logarithms
203
7° (187°)
(352°) 172°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.08 589
9.99 675
9.08 914
0.91 086
0.00 325
0.91 411
60
1
.08 692
.99 674
.09 019
.90 981
.00326
.91 308
59
2
.08 795
.99 672
.09 123
.90 877
.00328
.91 205
58
3
.08 897
.99 670
.09 227
.90 773
.00 330
.91 103
57
4
.08 999
.99 669
.09 330
.90 670
.00 331
.91 001
56
5
9.09 101
9.99 667
9.09 434
0.90 566
0.00 333
0.90 899
55
6
.09 202
.99 666
.09 537
.90 463
.00334
.90 798
54
7
.09 304
.99664
.09640
.90 360
.00 336
.90 696
53
8
.09 405
.99 663
.09 742
.90 258
.00337
.90 595
52
9
.09 506
.99 661
.09845
.90 155
.00 339
.90 494
51
10
9.09 606
9.99 659
9.09 947
0.90 053
0.00 341
0.90 394
50
11
.09 707
.99 658
.10049
.89 951
.00 342
.90 293
49
12
.09 807
.99 656
.10 150
.89 850
.00344
.90 193
48
13
.09907
.99 655
.10 252
.89 748
.00345
.90093
47
14
.10 006
.99 653
.10353
.89647
.00347
.89 994
46
15
9.10 106
9.99 651
9.10 454
0.89 546
0.00 349
0.89 894
45
16
.10 205
.99 650
.10 555
.89 445
.00350
.89 795
44
17
.10 304
.99648
.10 656
.89 344
.00 352
.89 696
43
18
.10 402
.99647
.10 756
.89 244
.00 353
.89 598
42
19
.10 501
.99645
.10 856
.89 144
.00 355
.89 499
41
20
9.10599
9.99 643
9.10 956
0.89044
0.00 357
0.89 401
40
21
.10 697
.99 642
.11 056
.88 944
.00 358
.89 303
39
22
.10 795
.99640
.11 155
.88845
.00360
.89 205
38
23
.10 893
.99 638
.11254
.88 746
.00362
.89 107
37
24
.10 990
.99 637
.11 353
.88647
.00 363
.89 010
36
25
9.11 087
9.99 635
9.11452
0.88 548
0.00 365
0.88913
35
26
.11 184
.99 633
.11 551
.88449
.00 367
.88 816
34
27
.11 281
.99 632
.11 649
.88351
.00368
.88 719
33
28
.11 377
.99 630
.11 747
.88253
.00 370
.88623
32
29
.11 474
.99 629
.11 845
.88155
.00371
.88 526
31
30
9.11 570
9.99 627
9.11 943
0.88 057
0.00 373
0.88 430
30
31
.11 666
.99 625
.12040
.87 960
.00375
.88 334
29
32
.11761
.99 624
.12 138
.87 862
.00376
.88239
28
33
.11 857
.99 622
.12 235
.87 765
.00378
.88 143
27
34
.11 952
.99 620
.12 332
.87 668
.00380
.88 048
26
35
9.12 047
9.99 618
9.12 428
0.87 572
0.00 382
0.87 953
25
36
.12 142
.99 617
.12 525
.87 475
.00 383
.87 858
24
37
.12 236
.99 615
.12 621
.87 379
.00 385
.87764
23
38
.12331
.99 613
.12717
.87283
.00387
.87 669
22
39
.12 425
.99 612
.12 813
.87 187
.00388
.87 575
21
40
9.12519
9.99 610
9.12 909
0.87 091
0.00 390
0.87 481
20
41
.12612
.99 608
.13004
.86 996
.00 392
.87 388
19
42
.12 706
.99 607
.13 099
.86 901
.00 393
.87 294
18
43
.12 799
.99 605
.13 194
.86806
.00395
.87 201
17
44
.12 892
.99 603
.13 289
.86 711
.00397
.87 108
16
45
9.12 985
9.99 601
9.13 384
0.86 616
0.00 399
0.87 015
15
46
.13 078
.99 600
.13 478
.86 522
.00 400
.86 922
14
47
.13 171
.99 598
.13 573
.86 427
.00402
.86 829
13
48
.13 263
.99 596
.13 667
.86 333
.00 404
.86 737
12
49
.13355
.99 595
.13 761
.86 239
.00 405
.86645
11
50
9.13447
9.99 593
9.13854
0.86 146
0.00 407
0.86 553
10
51
.13 539
.99 591
.13 948
.86 052
.00409
.86 461
9
52
.13 630
.99 589
.14041
.85959
.00411
.86 370
8
53
.13 722
.99 588
.14 134
.85866
.00412
.86 278
7
54
.13 813
.99 586
.14 227
.85 773
.00 414
.86 187
6
55
9.13 904
9.99 584
9.14 320
0.85 680
0.00416
0.86 096
5
56
.13 994
.99 582
.14412
.85588
.00 418
.86 006
4
57
.14 085
.99 581
.14 504
.85496
.00 419
.85 915
3
58
.14 175
.99 579
.14 597
.85 403
.00 421
.85 825
2
59
.14 266
.99 577
.14 688
.85312
.00 423
.85 734
1
60
9.14356
9.99 575
9.14 780
0.85 220
0.00 425
0.85 644
0
Cos
Sin
Cot
Tan
Csc
Sec
'
97° (277°)
(262°) 82°
204
Table 4. Trigonometric Logarithms
8° (188°)
(351°) 171°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.14356
9.99 575
9.14 780
0.85 220
0.00 425
0.85 644
60
1
.14445
.99 574
.14 872
.85 128
.00 426
.85 555
59
2
.14 535
.99 572
.14 963
.85 037
.00 428
.85465
58
3
.14 624
.99 570
.15 054
.84946
.00 430
.85 376
57
4
.14714
.99 568
.15 145
.84855
.00432
.85 286
56
5
9.14 803
9.99 566
9.15 236
0.84764
0.00 434
0.85 197
55
6
.14891
.99 565
.15327
.84 673
.00 435
.85 109
54
7
.14 980
.99 563
.15417
.84583
.00 437
.85 020
53
8
.15 069
.99 561
.15 508
.84492
.00 439
.84931
52
9
.15 157
.99 559
.15 598
.84402
.00 441
.84843
51
10
9.15 245
9.99 557
9.15 688
0.84 312
0.00 443
0.84 755
50
11
.15 333
.99 556
.15 777
.84223
.00 444
.84667
49
12
.15421
.99 554
.15 867
.84 133
.00 446
.84579
48
13
.15 508
.99 552
.15 956
.84044
.00 448
.84 492
47
14
.15 596
.99 550
.16 046
.83 954
.00 450
.84404
46
15
9.15 683
9.99 548
9.16 135
0.83 865
0.00 452
0.84 317
45
16
.15770
.99 546
.16 224
.83776
.00 454
.84230
44
17
.15 857
.99 545
.16312
.83 688
.00 455
.84 143
43
18
.15 944
.99 543
.16401
.83 599
.00457
.84056
42
19
.16 030
.99 541
.16489
.83511
.00 459
.83 970
41
20
9.16116
9.99 539
9.16 577
0.83 423
0.00 461
0.83 884
40
21
.16 203
.99 537
.16 665
.83 335
.00463
.83 797
39
22
.16 289
.99 535
.16753
.83 247
.00 465
.83711
38
23
.16374
.99 533
.16841
.83 159
.00 467
.83 626
37
24
.16 460
.99 532
.16928
.83072
.00 468
.83 540
36
25
9.16 545
9.99 530
9.17016
0.82 984
0.00 470
0.83 455
35
26
.16631
.99 528
.17 103
.82 897
.00 472
.83 369
34
27
.16716
.99 526
.17 190
.82 810
.00474
.83284
33
28
.16801
.99524
.17 277
.82 723
.00 476
.83 199
32
29
.16 886
.99 522
.17363
.82 637
.00 478
.83114
31
30
9.16 970
9.99 520
9.17 450
0.82 550
0.00 480
0.83 030
30
31
.17 055
.99 518
.17 536
.82 464
.00 482
.82 945
29
32
.17 139
.99 517
.17 622
.82 378
.00 483
.82 861
28
33
.17 223
.99 515
.17 708
.82 292
.00 485
.82 777
27
34
.17307
.99 513
.17 794
.82 206
.00 487
.82 693
26
35
9.17391
9.99511
9.17 880
0.82 120
0.00 489
0.82 609
25
36
.17474
.99 509
.17 965
.82 035
.00 491
.82 526
24
37
.17 558
.99 507
.18 051
.81 949
.00 493
.82 442
23
38
.17 641
.99 505
.18 136
.81864
.00 495
.82 359
22
39
.17 724
.99 503
.18221
.81 779
.00 497
.82 276
21
40
9.17 807
9.99 501
9.18 306
0.81 694
0.00 499
0.82 193
20
41
.17 890
.99 499
.18391
.81 609
.00 501
.82 110
19
42
.17 973
.99 497
.18 475
.81 525
.00 503
.82 027
18
43
.18 055
.99 495
.18 560
.81 440
.00505
.81 945
17
44
.18 137
.99 494
.18 644
.81 356
.00 506
.81 863
16
45
9.18 220
9.99 492
9.18728
0.81 272
0.00 508
0.81 780
15
46
.18 302
.99 490
.18812
.81 188
.00 510
.81 698
14
47
.18383
.99 488
.18 896
.81 104
.00 512
.81 617
13
48 •
.18465
.99 486
.18 979
.81 021
.00 514
.81 535
12
49
.18 547
.99484
.19 063
.80 937
.00 516
.81 453
11
50
9.18 628
9.99 482
9.19 146
0.80 854
0.00 518
0.81 372
10
51
.18 709
.99 480
.19 229
.80 771
.00520
.81 291
9
52
.18790
.99 478
.19312
.80 688
.00 522
.81 210
8
53
.18871
.99 476
.19 395
.80 605
.00 524
.81 129
7
54
.18 952
.99 474
.19 478
.80522
.00526
.81 048
6
55
9.19 033
9.99 472
9.19 561
0.80 439
0.00 528
0.80 967
5
56
.19 113
.99 470
.19 643
.80 357
.00 530
.80 887
4
57
.19 193
.99 468
.19 725
.80 275
.00532
.80 807
3
58
.19 273
.99 466
.19807
.80 193
.00 534
.80 727
2
59
.19 353
.99464
.19 889
.80 111
.00536
.80 647
1
60
9.19433
9.99 462
9.19971
0.80 029
0.00 538
0.80 567
0
Cos
Sin
Cot
Tan
Csc
Sec
'
98° (278°)
(261°) 81°
Table 4. Trigonometric Logarithms
205
9° (189°)
(350°) 170°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.19 433
9.99 462
9.19971
0.80 029
0.00 538
0.80 567
60
1
.19513
.99 460
.20 053
.79 947
.00540
.80 487
59
2
.19 592
.99 458
.20 134
.79 866
.00542
.80 408
58
3
.19 672
.99 456
.20 216
.79784
.00544
.80 328
57
4
.19 751
.99 454
.20 297
.79 703
.00 546
.80249
56
5
9.19 830
9.99 452
9.20 378
0.79 622
0.00 548
0.80 170
55
6
.19 909
.99 450
.20 459
.79541
.00550
.80 091
54
7
.19 988
.99 448
.20 540
.79 460
.00552
.80 012
53
8
.20 067
.99 446
.20 621
.79 379
.00554
.79 933
52
9
.20 145
.99444
.20 701
.79 299
.00556
.79 855
51
10
9.20 223
9.99 442
9.20 782
0.79 218
0.00 558
0.79 777
50
11
.20 302
.99 440
.20 862
.79 138
.00560
.79 698
49
12
.20 380
.99 438
.20 942
.79 058
.00562
.79 620
48
13
.20 458
.99436
.21 022
.78 978
.00564
.79 542
47
14
.20 535
.99 434
.21 102
.78 898
.00566
.79 465
46
15
9.20 613
9.99 432
9.21 182
0.78 818
0.00 568
0.79 387
45
16
.20 691
.99 429
.21 261
.78 739
.00 571
.79 309
44
17
.20 768
.99 427
.21 341
.78 659
.00573
.79 232
43
18
.20845
.99 425
.21 420
.78 580
.00 575
.79 155
42
19
.20 922
.99 423
.21 499
.78 501
.00577
.79 078
41
20
9.20 999
9.99 421
9.21 578
0.78 422
0.00 579
0.79 001
40
21
.21 076
.99 419
.21 657
.78 343
.00581
.78 924
39
22
.21 153
.99417
.21 736
.78 264
.00583
.78 847
38
23
.21 229
.99 415
.21 814
.78 186
.00 585
.78 771
37
24
.21 306
.99 413
.21 893
.78 107
.00 587
.78 694
36
25
9.21 382
9.99411
9.21 971
0.78 029
0.00 589
0.78 618
35
26
.21 458
.99 409
.22 049
.77 951
.00 591
.78 542
34
27
.21534
.99 407
.22 127
.77 873
.00593
-.78466
33
28
.21 610
.99404
.22 205
.77 795
.00596
.78 390
32
29
.21 685
.99 402
.22 283
.77 717
.00598
.78 315
31
30
9.21 761
9.99 400
9.22 361
0.77 639
0.00 600
0.78 239
30
31
.21 836
.99 398
.22 438
.77 562
.00602
.78164
29
32
.21 912
.99 396
.22 516
.77 484
.00604
.78 088
28
33
.21 987
.99394
.22 593
.77 407
.00 606
.78 013
27
34
.22 062
.99 392
.22 670
.77 330
.00608
.77 938
26
35
9.22 137
9.99 390
9.22 747
0.77 253
0.00 610
0.77 863
25
36
.22211
.99 388
.22 824
.77 176
.00612
.77 789
24
37
.22 286
.99 385
.22 901
.77 099
.00615
.77 714
23
38
.22 361
.99383
.22 977
.77 023
.00 617
.77 639
22
39
.22 435
.99 381
.23054
.76 946
.00619
.77 565
21
40
9.22 509
9.99 379
9.23 130
0.76 870
0.00 621
0.77 491
20
41
.22 583
.99 377
.23 206
.76 794
.00 623
.77 417
19
42
.22 657
.99 375
.23 283
.76 717
.00625
.77 343
18
43
.22 731
.99 372
.23 359
.76641
.00628
.77 269
17
44
.22 805
.99 370
.23 435
.76 565
.00630
.77 195
16
45
9.22 878
9.99 368
9.23 510
0.76 490
0.00 632
0.77 122
15
46
.22 952
.99 366
.23 586
.76 414
.00634
.77048
14
47
.23 025
.99364
.23 661
.76 339
.00636
.76 975
13
48
.23 098
.99 362
.23 737
.76 263
.00638
.76 902
12
49
.23 171
.99 359
.23 812
.76 188
.00641
.76 829
11
50
9.23 244
9.99 357
9.23 887
0.76 113
0.00 643
0.76 756
10
51
.23 317
.99 355
.23 962
.76 038
.00645
.76683
9
52
.23 390
.99 353
.24 037
.75 963
.00647
.76 610
8
53
.23 462
.99 351
.24 112
.75 888
.00649
.76 538
7
54
.23 535
.99 348
.24 186
.75 814
.00652
.76 465
6
55
9.23 607
9.99 346
9.24 261
0.75 739
0.00654
0.76 393
5
56
.23 679
.99 344
.24 335
.75 665
.00656
.76 321
4
57
.23 752
.99342
.24 410
.75 590
.00 658
.76 248
3
58
.23 823
.99 340
.24484
.75 516
.00 660
.76 177
2
59
.23 895
.99 337
.24 558
.75 442
.00 663
.76 105
1
60
9.23 967
9.99 335
9.24 632
0.75 368
0.00 665
0.76 033
0
Cos
Sin
Cot
Tail
Csc
Sec
'
99° (279°)
(260°) 80°
206
Table 4. Trigonometric Logarithms
10° (190°)
(349°) 169°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.23 967
9.99 335
9.24 632
0.75 368
0.00 665
0.76 033
60
1
.24 039
.99 333
.24 706
.75 294
.00 667
.75 961
59
2
.24 110
.99 331
.24 779
.75 221
.00 669
.75 890
58
3
.24 181
.99 328
.24 853
.75 147
.00 672
.75 819
57
4
.24 253
.99 326
.24 926
.75 074
.00 674
.75 747
56
5
9.24 324
9.99 324
9.25 000
0.75 000
0.00 676
0.75 676
55
6
.24 395
.99 322
.25 073
.74 927
.00 678
.75 605
54
7
.24 466
.99 319
.25 146
.74 854
.00 681
.75 534
53
8
.24 536
.99317
.25 219
.74 781
.00683
.75 464
52
9
.24 607
.99 315
.25 292
.74 708
.00 685
.75 393
51
10
9.24 677
9.99 313
9.25 365
0.74 635
0.00 687
0.75 323
50
11
.24 748
.99 310
.25 437
.74 563
.00 690
.75 252
49
12
.24 818
.99 308
.25 510
.74 490
.00692
.75 182
48
13
.24 888
.99 306
.25 582
.74 418
.00 694
.75 112
47
14
.24 958
.99 304
.25 655
.74 345
.00696
.75 042
46
15
9.25 028
9.99 301
9.25 727
0.74 273
0.00 699
0.74 972
45
16
.25 098
.99 299
.25 799
.74 201
.00 701
.74 902
44
17
.25 168
.99 297
.25 871
.74 129
.00 703
.74 832
43
18
.25 237
.99 294
.25 943
.74 057
.00 706
.74 763
42
19
.25 307
.99 292
.26 015
.73 985
.00 708
.74 693
41
20
9.25 376
9.99 290
9.26 086
0.73 914
0.00 710
0.74 624
40
21
.25 445
.99 288
.26 158
.73842
.00712
.74 555
39
22
.25 514
.99 285
.26 229
.73 771
.00 715
.74 486
38
23
.25 583
.99 283
.26 301
.73 699
.00 717
.74 417
37
24
.25 652
.99 281
.26 372
.73 628
.00719
.74 348
36
25
9.25 721
9.99 278
9.26 443
0.73 557
0.00 722
0.74 279
35
26
.25 790
.99 276
.26 514
.73 486
.00 724
.74 210
34
27
.25 858
.99 274
.26 585
.73 415
.00 726
.74 142
33
28
.25 927
.99 271
.26 655
.73 345
.00 729
.74 073
32
29
.25 995
.99 269
.26 726
.73 274
.00 731
.74 005
31
30
9.26 063
9.99 267
9.26 797
0.73 203
0.00 733
0.73 937
30
31
.26 131
.99 264
.26 867
.73 133
.00 736
.73 869
29
32
.26 199
.99 262
.26 937
.73 063
.00 738
.73 801
28
33
.26 267
.99 260
.27 008
.72 992
• .00740
.73 733
27
34
.26 335
.99 257
.27 078
.72 922
.00 743
.73 665
26
35
9.26 403
9.99 255
9.27 148
0.72 852
0.00 745
0.73 597
25
36
.26 470
.99 252
.27 218
.72 782
.00 748
.73 530
24
37
.26 538
.99 250
.27 288
.72 712
.00 750
.73 462
23
38
.26 605
.99 248
.27 357
.72 643
.00 752
.73 395
22
39
.26 672
.99 245
.27 427
.72 573
.00 755
.73 328
21
40
9.26 739
9.99 243
9.27 496
0.72 504
0.00 757
0.73 261
20
41
.26 806
.99 241
.27 566
.72 434
.00 759
.73 194
19
42
.26 873
.99 238
.27 635
.72 365
.00 762
.73 127
18
43
.26 940
.99 236
.27 704
.72 296
.00 764
.73 060
17
44
.27 007
.99 233
.27 773
.72 227
.00 767
.72 993
16
45
9.27 073
9.99 231
9.27 842
0.72 158
0.00 769
0.72 927
15
46
.27 140
.99 229
.27911
.72 089
.00 771
.*2 860
14
47
.27 206
.99 226
.27 980
.72 020
.00 774
.72 794
13
48
.27 273
.99 224
.28 049
.71 951
.00776
.72 727
12
49
.27 339
.99 221
.28 117
.71 883
.00 779
.72 661
11
50
9.'27 405
9.99 219
9.28 186
0.71 814
0.00 781
0.72 595
10
51
.27 471
.99 217
.28 254
.71 746
.00 783
.72 529
9
52
.27 537
.99 214
.28 323
.71 677
.00 786
.72 463
8
53
.27 602
.99 212
.28 391
.71 609
.00 788
.72 398
7
54
.27 668
.99 209
.28 459
.71 541
.00 791
.72 332
6
55
9.27 734
9.99 207
9.28 527
0.71 473
0.00 793
0.72 266
5
56
.27 799
.99 204
.28 595
.71 405
.00 796
.72 201
4
57
.27 864
.99 202
.28 662
.71 338
.00 798
.72 136
3
58
.27 930
.99 200
.28 730
.71 270
.00 800
.72 070
2
59
.27 995
.99 197
.28 798
.71 202
.00 803
.72 005
1
60
9.28 060
9.99 195
9.28 865
0.71 135
0.00 805
0.71 940
0
Cos
Sin
Cot
Tan
Csc
Sec
'
100° (280°)
(259°) 79°
Table 4. Trigonometric Logarithms
207
11° (191°)
(348°) 168°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.28 060
9.99 195
9.28 865
0.71 135
0.00 805
0.71 940
60
1
.28 125
.99 192
.28 933
.71 067
.00808
.71 875
59
2
.28 190
.99 190
.29 000
.71 000
.00810
.71 810
58
3
.28 254
.99 187
.29 067
.70 933
.00813
.71 746
57
4
.28 319
.99185
.29 134
.70 866
.00815
.71 681
56
5
9.28 384
9.99 182
9.29 201
0.70 799
0.00 818
0.71 616
55
6
.28448
.99 180
.29 268
.70 732
.00820
.71 552
54
7
.28 512
.99 177
.29 335
.70 665
.00823
.71 488
53
8
.28 577
.99 175
.29 402
.70 598
.00825
.71 423
52
9
.28641
.99 172
.29 468
.70 532
.00 828
.71 359
51
10
9.28 705
9.99 170
9.29 535
0.70 465
0.00 830
0.71 295
50
11
.28 769
.99 167
.29 601
.70 399
.00833
.71 231
49
12
.28833
.99 165
.29 668
.70 332
.00835
.71 167
48
13
.28 896
.99 162
.29 734
.70 266
.00838
.71 104
47
14
.28 960
.99 160
.29 800
.70 200
.00840
.71040
46
15
9.29 024
9.99 157
9.29 866
0.70 134
0.00 843
0.70 976
45
16
.29 087
.99 155
.29 932
.70 068
.00845
.70 913
44
17
.29 150
.99 152
.29 998
.70 002
.00848
.70 850
43
18
.29 214
.99 150
.30 064
.69 936
.00 850
.70 786
42
19
.29 277
.99 147
.30 130
.69 870
.00853
.70 723
41
20
9.29 340
9.99 145
9.30 195
0.69 805
0.00 855
0.70 660
40
21
.29 403
.99 142
.30 261
.69 739
.00858
.70 597
39
22
.29 466
.99 140
.30 326
.69 674
.00 860
. .70534
38
23
.29 529
.99 137
.30 391
.69 609
.00863
.70 471
37
24
.29 591
.99 135
.30 457
.69543
.00865
.70 409
36
25
9.29 654
9.99 132
9.30 522
0.69 478
0.00 868
0.70 346
35
26
.29 716
.99 130
.30 587
.69 413
.00870
.70284
34
27
.29 779
.99 127
.30 652
.69 348
.00873
.70 221
33
28
.29841
.99 124
.30 717
.69 283
.00876
.70 159
32
29
.29 903
.99 122
. .30 782
.69 218
.00878
.70 097
31
30
9.29 966
9.99 119
9.30 846
0.69 154
0.00 881
0.70 034
30
31
.30 028
.99 fl7
.30911
.69 089
.00883
.69 972
29
32
.30 090
.99 114
.30 975
.69 025
.00886
.69 910
28
33
.30 151
.99 112
.31 040
.68 960
.00888
.69849
27
34
.30 213
.99 109
.31 104
.68 896
.00 891
.69 787
26
35
9.30 275
9.99 106
9.31 168
0.68 832
0.00 894
0.69 725
25
36
.30 336
.99 104
.31 233
.68 767
.00 896
.69 664
24
37
.30 398
.99 101
.31 297
.68 703
.00 899
.69 602
23
38
.30 459
.99 099
.31 361
.68 639
.00 901
.69 541
22
39
.30 521
.99 096
.31 425
.68 575
.00904
.69 479
21
40
9.30 582
9.99 093
9.31 489
0.68 511
0.00 907
0.69 418
20
41
.30643
.99 091
.31 552
.68 448
.00909
.69 357
19
42
.30704
.99088
.31 616
.68384
.00 912
.69 296
18
43
.30 765
.99 086
.31 679
.68 321
.00914
.69 235
17
44
.30 826
.99 083
.31 743
.68 257
.00917
.69 174
16
45
9.30 887
9.99 080
9.31 806
0.68 194
0.00 920
0.69 113
15
46
.30 947
.99 078
.31 870
.68 130
.00922
.69 053
14
47
.31 008
.99 075
.31 933
.68 067
.00925
.68 992
13
48
.31 068
.99 072
.31 996
.68004
.00928
.68 932
12
49
.31 129
.99 070
.32 059
.67 941
.00930
.68 871
11
50
9.31 189
9.99 067
9.32 122
0.67 878
0.00 933
0.68 811
10
51
.31 250
.99 064
.32 185
.67 815
.00936
.68 750
9
52
.31 310
.99 062
.32 248
.67 752
.00938
.68 690
8
53
.31 370
.99 059
.32311
.67 689
.00941
.68 630
7
54
.31 430
.99 056
.32 373
.67 627
.00944
.68570
6
55
9.31 490
9.99 054
9.32 436
0.67 564
0.00 946
0.68 510
5
56
.31 549
.99 051
.32 498
.67 502
.00949
.68 451
4
57
.31 609
.99048
.32 561
.67 439
.00952
.68 391
3
58
.31 669
.99046
.32 623
.67 377
.00954
.68 331
2
59
.31 728
.99043
.32685
.67 315
.00957
.68272
1
60
9.31 788
9.99 040
9.32 747
0.67 253
0.00960
0.68 212
0
Cos
Sin
Cot
Tan
Csc
Sec
'
101° (281°)
(258°) 78°
208
Table 4. Trigonometric Logarithms
12° (192°)
(347°) 167 c
'
Sin
Cos
Tan
Cot
Sec
CM-
0
9.31 788
9.99 040
9.32 747
0.67 253
0.00 960
0.68 212
60
1
.31 847
.99 038
.32 810
.67 190
.00 962
.68 153
59
2
.31 907
.99 035
.32 872
.67 128
.00 965
.68 093
58
3
.31 966
.99 032
.32 933
.67 067
.00 968
.68 034
57
4
.32 025
.99 030
.32 995
.67 005
.00 970
.67 975
56
5
9.32 084
9.99 027
9.33 057
0.66 943
0.00 973
0.67 916
55
6
.32 143
.99 024
.33 119
.66 881
.00 976
.67 857
54
7
.32 202
.99 022
.33 180
.66 820
.00 978
.67 798
53
8
.32 261
.99 019
.33 242
.66 758
.00 981
.67 739
52
9
.32 319
.99 016
.33 303
.66 697
.00984
.67 681
51
10
9.32 378
9.99 013
9.33 365
0.66 635
0.00 987
0.67 622
50
11
.32 437
.99 Oil
.33 426
.66 574
.00 989
.67 563
49
12
.32 495
.99 008
.33 487
.66 513
.00 992
.67 505
48
13
.32 553
.99 005
.33 548
.66 452
.00 995
.67 447
47
14
.32 612
.99 002
.33 609
.66 391
.00 998
.67 388
46
15
9.32 670
9.99 000
9.33 670
0.66 330
0.01 000
0.67 330
45
16
.32 728
.98 997
.33 731
.66 269
.01 003
.67 272
44
17
.32 786
.98 994
.33 792
.66 208
.01 006
.67 214
43
18
.32844
.98 991
.33 853
.66 147
.01 009
.67 156
42
19
.32 902
.98 989
.33 913
.66 087
.01 Oil
.67 098
41
20
9.32 960
9.98 986
9.33 974
0.66 026
0.01 014
0.67 040
40
21
.33 018
.98 983
.34 034
.65 966
.01 017
.66 982
39
22
.33 075
.98 980
.34 095
.65 905
.01 020
.66 925
38
23
.33 133
.98 978
.34 155
.65845
.01 022
.66 867
37
24
.33 190
.98 975
.34 215
.65 785
.01 025
.66 810
36
25
9.33 248
9.98 972
9.34 276
0.65 724
0.01 028
0.66 752
35
26
.33 305
• .98969
.34 336
.65 664
.01 031
.66 695
34
27
.33 362
.98 967
.34 396
.65604
.01 033
.66 638
33
28
.33 420
.98 964
.34 456
.65544
.01 036
.66 580
32
29
.33 477
.98 961
.34 516
.65484
.01 039
.66 523
31
30
9.33 534
9.98 958
9.34 576
0.65 424
0.01 042
0.66 466
30
31
.33 591
.98 955
.34 635
.65 365
*.01 045
.66 409
29
32
.33 647
.98 953
.34 695
.65 305
.01 047
.66 353
28
33
.33 704
.98 950
.34 755
.65 245
.01 050
.66 296
27
34
.33 761
.98 947
.34 814
.65 186
.01 053
.66 239
26
35
9.33 818
9.98 944
9.34 874
0.65 126
0.01 056
0.66 182
25
36
.33 874
.98 941
.34 933
.65 067
.01 059
.66 126
24
37
.33 931
.98 938
.34 992
.65 008
.01 062
.66 069
23
38
.33 987
.98 936
.35 051
.64 949
.01 064
.66 013
22
39
.34 043
.98 933
.35 111
.64 889
.01 067
.65 957
21
40
9.34 100
9.98 930
9.35 170
0.64 830
0.01 070
0.65 900
20
41
.34 156
.98 927
.35 229
.64771
.01 073
.65 844
19
42
.34 212
.98 924
.35 288
.64712
.01 076
.65 788
18
43
.34 268
.98 921
.35 347
.64 653
.01 079
.65 732
17
44
.34 324
.98 919
.35 405
.64595
.01 081
.65 676
16
45
9.34 380
9.98 916
9.35 464
0.64 536
0.01 084
0.65 620
15
46
.34 436
.98 913
.35 523
.64477
.01 087
.65 564
14
47
.34 491
.98 910
.35 581
.64419
.01 090
65509
13
48
.34 547
.98 907
.35 640
.64360
.01 093
.65 453
12
49
.34 602
.98 904
.35 698
.64302
.01 096
.65 398
11
50
9.34 658
9.98 901
9.35 757
0.64 243
0.01 099
0.65 342
10
51
.34 713
.98 898
.35 815
.64185
.01 102
.65 287
9
52
.34 769
.98 896
.35 873
.64 127
.01 104
.65 231
8
53
.34 824
.98 893
.35 931
.64069
.01 107
.65 176
7
54
.34 879
.98 890
.35 989
.64011
.01 110
.65 121
6
55
9.34 934
9.98 887
9.36 047
0.63 953
0.01 113
0.65 066
5
56
.34 989
.98884
.36 105
.63 895
.01 116
.65011
4
57
.35 044
.98 881
.36 163
.63 837
.01 119
.64956
3
58
.35 099
.98 878
.36 221
.63 779
.01 122
.64 901
2
59
.35 154
.98 875
.36 279
.63 721
.01 125
.64 846
1
60
9.35 209
9.98 872
9.36 336
0.63 664
0.01 128
0.64 791
0
Cos
Sin
Cot
Tan
Csc
Sec
'
102° (282°)
(257°) 77C
Table 4. Trigonometric Logarithms
209
13° (193°)
(346°) 166°
'
Sin
Cos
Tan Cot
Sec
Csc
0
9.35 209
9.98 872
9.36 336
0.63 664
0.01 128
0.64 791
60
1
.35 263
.98 869
.36 394
.63 606
.01 131
.64737
59
2
.35 318
.98 867
.36 452
.63 548
.01 133
.64 682
58
3
.35 373
.98864
.36 509
.63 491
.01 136
.64 627
57
4
.35 427
.98 861
.36 566
.63 434
.01 139
.64 573
56
5
9.35 481
9.98 858
9.36 624
0.63 376
0.01 142
0.64 519
55
6
.35 536
.98855
.36 681
.63 319
.01 145
.64464
54
7
.35 590
.98852
.36 738
.63 262
.01 148
.64410
53
8
.35 644
.98849
.36 795
.63 205
.01 151
.64356
52
9
.35 698
.98846
.36 852
.63 148
.01 154
.64302
51
10
9.35 752
9.98 843
9.36 909
0.63 091
0.01 157
0.64 248
50
11
.35 806
.98840
.36 966
.63 034
.01 160
.64 194
49
12
.35 860
.98 837
.37 023
.62 977
.01 163
.64 140
48
13
.35 914
.98 834
.37 080
.62 920
.01 166
.64 086
47
14
.35 968
.98 831
.37 137
.62 863
.01 169
.64032
46
15
9.36 022
9.98 828
9.37 193
0.62 807
0.01 172
0.63 978
45
16
.36 075
.98 825
.37 250
.62 750
.01 175
.63 925
44
17
.36 129
.98 822
.37 306
.62 694
.01 178
.63 871
43
18
.36 182
.98 819
.37 363
.62 637
.01 181
.63 818
42
19
.36 236
.98 816
.37 419
.62 581
.01 184
.63 764
41
20
9.36 289
9.98 813
9.37 476
0.62 524
0.01 187
0.63 711
40
21
.36 342
.98 810
.37 532
.62 468
.01 190
.63 658
39
22
.36 395
.98 807
.37 588
.62 412
.01 193
.63 605
38
23
.36 449
.98804
.37644
.62 356
.01 196
.63 551
37
24
.36 502
.98 801
.37 700
.62 300
.01 199
.63 498
36
25
9.36 555
9.98 798
9.37 756
0.62 244
0.01 202
0.63 445
35
26
.36 608
.98 795
.37 812
.62 188
.01 205
.63 392
34
27
.36 660
.98 792
.37 868
.62 132
.01 208
.63 340
33
28
.36 713
.98 789
.37 924
.62 076
.01 211
.63 287
32
29
.36 766
.98 786
.37980
.62 020
.01 214
.63 234
31
30
9.36 819
9.98 783
9.38 035
0.61 965
0.01 217
0.63 181
30
31
.36 871
.98 780
.38 091
.61 909
.01 220
.63 129
29
32
.36 924
.98 777
.38 147
.61 853
.01 223
.63 076
28
33
.36 976
.98 774
.38202
.61 798
.01 226
.63 024
27
34
.37 028
.98 771
.38 257
.61 743
.01 229
.62 972
26
35
9.37 081
9.98 768
9.38 313
0.61 687
0.01 232
0.62 919
25
36
.37 133
.98 765
.38368
.61 632
.01 235
.62 867
24
37
.37 185
.98 762
.38 423
.61 577
.01 238
.62 815
23
38
.37 237
.98 759
.38 479
.61 521
.01 241
.62 763
22
39
.37 289
.98 756
.38 534
.61 466
.01 244
.62711
21
40
9.37 341
9.98 753
9.38 589
0.61 411
0.01 247
0.62 659
20
41
.37 393
.98 750
.38644
.61 356
.01 250
.62 607
19
42
.37445
.98 746
.38 699
.61 301
.01 254
.62 555
18
43
.37 497
.98 743
.38754
.61 246
.01 257
.62 503
17
44
.37 549
.98 740
.38 808
.61 192
.01 260
.62 451
16
45
9.37 600
9.98 737
9.38 863
0.61 137
0.01 263
0.62 400
15
46
.37 652
.98 734
.38 918
.61 082
.01 266
.62 348
14
47
.37 703
.98 731
.38 972
.61 028
.01 269
.62 297
13
48
.37 755
.98 728
.39 027
.60 973
.01 272
.62 245
12
49
.37806
.98 725
.39 082
.60 918
.01 275
.62 194
11
50
9.37 858
9.98 722
9.39 136
0.60 864
0.01 278
0.62 142
10
51
.37909
.98 719
.39 190
.60 810
.01 281
.62 091
9
52
.37 960
.98 715
.39 245
.60 755
.01 285
.62040
8
53
.38011
.98 712
.39 299
.60 701
.01 288
.61 989
7
54
.38 062
.98 709
.39 353
.60647
.01 291
.61 938
6
55
9.38 113
9.98 706
9.39 407
0.60 593
0.01 294
0.61 887
5
56
.38 164
.98 703
.39 461
.60 539
.01 297
.61836
4
57
.38 215
.98700
.39 515
.60 485
.01 300
.61 785
3
58
.38 266
.98 697
.39 569
.60431
.01 303
.61 734
2
59
.38 317
.98 694
.39 623
.60 377
.01 306
.61683
1
60
9.38 368
9.98 690
9.39 677
0.60 323
0.01 310
0.61 632
0
Cos Sin
Cot
Tan
Csc
GAA
'
103° (283°)
(256°) 76°
210
Table 4. Trigonometric Logarithms
14° (194°)
(345°) 165°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.38 368
9.98 690
9.39 677
0.60 323
0.01 310
0.61 632
60
1
.38418
.98 687
.39 731
.60 269
.01 313
.61 582
59
2
.38 469
.98 684
.39 785
.60 215
.01 316
.61 531
58
3
.38519
.98 681
.39 838
.60 162
.01 319
.61 481
57
4
.38 570
.98 678
.39 892
.60 108
.01 322
.61 430
56
5
9.38 620
9.98 675
9.39 945
0.60 055
0.01 325
0.61 380
55
6
.38 670
.98 671
.39 999
.60 001
.01 329
.61 330
54
7
.38 721
.98 668
.40 052
.59 948
.01 332
.61 279
53
8
.38 771
.98 665
.40 106
.59 894
.01 335
.61 229
52
9
.38 821
.98 662
.40 159
.59841
.01 338
.61 179
51
10
9.38 871
9.98 659
9.40 212
0.59 788
0.01 341
0.61 129
50
11
.38 921
.98 656
.40 266
.59 734
.01 344
.61 079
49
12
.38 971
.98 652
.40 319
.59 681
.01 348
.61 029
48
13
.39 021
.98 649
.40 372
.59 628
.01 351
.60 979
47
14
.39 071
.98 646
.40 425
.59 575
.01 354
.60 929
46
15
9.39 121
9.98 643
9.40 478
0.59 522
0.01 357
0.60 879
45
16
.39 170
.98 640
.40 531
.59 469
.01 360
.60 830
44
17
.39 220
.98 636
.40 584
.59 416
.01 364
.60 780
43
18
.39 270
.98 633
.40 636
.59364
.01 367
.60 730
42
19
.39 319
.98 630
.40 689
.59 311
.01 370
.60 681
41
20
9.39 369
9.98 627
9.40 742
0.59 258
0.01 373
0.60 631
40
21
.39 418
.98 623
.40 795
.59 205
.01 377
.60 582
39
22
.39 467
.98 620
.40 847
.59 153
.01 380
.60 533
38
23
.39 517
.98 617
.40 900
.59 100
.01 383
.60 483
37
24
.39 566
.98 614
.40 952
.59 048
.01 386
.60 434
36
25
9.39 615
9.98610
9.41 005
0.58 995
0.01 390
0.60 385
35
26
.39 664
.98 607
.41 057
.58 943
.01 393
.60 336
34
27
.39 713
.98 604
.41 109
.58 891
.01 396
.60 287
33
28
.39 762
.98 601
.41 161
.58 839
.01 399
.60 238
32
29
.39 811
.98 597
.41 214
.58 786
.01 403
.60 189
31
30
9.39 860
9.98 594
9.41 266
0.58 734
0.01 406
0.60 140
30
31
.39 909
.98 591
.41 318
.58 682
.01 409
.60 091
29
32
.39 958
.98 588
.41 370
.58 630
.01 412
.60 042
28
33
.40 006
.98 584
.41 422
.58 578
.01 416
.59 994
27
34
.40 055
.98 581
.41 474
.58 526
.01 419
.59 945
26
35
9.40 103
9.98 578
9.41 526
0.58 474
0.01 422
0.59 897
25
36
.40 152
.98 574
.41 578
.58 422
.01 426
.59848
24
37
.40 200
.98 571
.41 629
.58 371
.01 429
.59 800
23
38
.40 249
.98 568
.41 681
.58319
.01 432
.59 751
22
39
.40 297
.98 565
.41 733
.58 267
.01 435
.59 703
21
40
9.40 346
9.98 561
9.41 784
0.58 216
0.01 439
0 59 654
20
41
.40 394
.98 558
.41 836
.58 164
.01 442
.59 606
19
42
.40 442'
.98 555
.41 887
.58 113
.01 445
.59 558
18
43
.40 490
.98 551
.41 939
.58 061
.01 449
.59 510
17
44
.40 538
.98 548
.41 990
.58 010
.01 452
.59 462
16
45
9.40 586
9.98 545
9.42 041
0.57 959
0.01 455
0.59 414
15
46
.40 634
.98 541
.42 093
.57 907
.01 459
.59 366
14
47
.40 682
.98 538
.42 144
.57 856
.01 462
,69318
13
48
.40 730
.98 535
.42 195
.57 805
.01465
.59 270
12
49
.40 778
.98 531
.42 246
.57 754
.01 469
.59 222
11
50
9.40 825
9.98 528
9.42 297
0.57 703
0.01 472
0.59 175
10
51
.40 873
.98 525
.42 348
.57 652
.01 475
.59 127
9
52
.40 921
.98 521
.42 399
.57 601
.01 479
.59 079
8
53
.40 968
.98518
.42 450
.57 550
.01 482
.59 032
7
54
,41 016
.98 515
.42 501
.57 499
.01 485
.58 984
6
55
9.41 063
9.98511
9.42 552
0.57 448
0.01 489
0.58 937
5
56
.41 111
.98 508
.42 603
.57 397
.01 492
.58 889
4
57
.41 158
.98 505
.42 653
.57 347
.01 495
.58 842
3
58
.41 205
.98 501
.42 704
.57 296
.01 499
.58 795
2
59
.41 252
.98 498
.42 755
.57 245
.01 502
.58 748
1
60
9.41 300
9.98 494
9.42 805
0.57 195
0.01 506
0.58 700
0
Cos
Sin
Cot
Tan
Csc
Sec
'
104° (284°)
(255°) 75°
Table 4. Trigonometric Logarithms
211
15° (195°)
(344°) 164°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.41 300
9.98 494
9.42 805
0.57 195
0.01 506
0.58 700
60
1
.41 347
.98 491
.42856
.57 144
.01 509
.58 653
59
2
.41 394
.US 4SS
.42906
.57 094
.01 512
.58606
58
3
.41 441
.98484
.42 957
.57043
.01 516
.58 559
57
4
.41488
.98 481
.43007
.56 993
.01 519
.58 512
56
5
9.41 535
9.98 477
9.43 057
0.56 943
0.01 523
0.58 465
55
6
.41 582
.98 474
.43 108
.56 892
.01 526
.58418
54
7
.41 628
.98 471
.43 158
.56842
.01 529
.58 372
53
8
.41 675
.98 467
.43 208
.56 792
.01 533
.58 325
52
9
.41 722
.98464
.43 258
.56 742
.01 536
.58 278
51
10
9.41 768
9.98 460
9.43 308
0.56 692
0.01 540
0.58 232
50
11
.41 815
.98 457
.43 358
.56 642
.01543
.58 185
49
12
.41 861
.98 453
.43 408
.56 592
.01 547
.58 139
48
13
.41 908
.98 450
.43 458
.56 542
.01 550
.58 092
47
14
.41 954
.98447
.43 508
.56 492
.01 553
.58 046
46
15
9.42 001
9.98 443
9.43 558
0.56 442
0.01 557
0.57 999
45
16
.42047
.98 440
.43 607
.56 393
.01 560
.57 953
44
17
.42 093
.98 436
.43 657
.56 343
.01 564
.57 907
43
18
.42 140
.98 433
.43 707
.56 293
.01 567
.57 860
42
19
.42 186
.98 429
.43 756
.56244
.01 571
.57 814
41
20
9.42 232
9.98 426
9.43 806
0.56 194
0.01 574
0.57 768
40
21
.42 278
.98 422
.43855
.56 145
.01 578
.57 722
39
22
.42 324
.98 419
.43 905
.56 095
.01 581
.57 676
38
23
.42 370
.98 415
.43 954
.56046
.01 585
.57 630
37
24
.42 416
.98 412
.44004
.55 996
.01 588
.57584
36
25
9.42 461
9.98 409
9.44 053
0.55 947
0.01 591
0.57 539
35
26
.42 507
.98 405
.44 102
.55 898
.01 595
.57 493
34
27
.42 553
.98 402
.44 151
.55849
.01 598
.57 447
33
28
.42 599
.98 398
.44201
.55 799
.01 602
.57 401
32
29
.42644
.98 395
.44250
.55 750
.01 605
.57 356
31
30
9.42 690
9.98 391
9.44 299
0.55 701
0.01 609
0.57 310
30
31
.42 735
.98 388
.44 348
.55 652
.01 612
.57 265
29
32
.42 781
.98384
.44397
.55 603
.01 616
.57 219
28
33
.42 826
.98 381
.44446
.55554
.01 619
.57 174
27
34
.42 872
.98 377
.44 495
.55 505
.01 623
.57 128
26
35
9.42 917
9.98 373
9.44 544
0.55 456
0.01 627
0.57 083
25
36
.42 962
.98 370
.44592
.55 408
.01 630
.57 038
24
37
.43008
.98 366
.44641
.55 359
.01 634
.56 992
23
38
.43 053
.98 363
.44690
.55 310
.01 637
.56 947
22
39
.43 098
.98 359
.44738
.55 262
.01 641
.56 902
21
40
9.43 143
9.98 356
9.44 787
0.55 213
0.01 644
0.56 857
20
41
.43 188
.98 352
.44836
.55 164
.01648
.56 812
19
42
.43 233
.98 349
.44884
.55 116
.01 651
.56 767
18
43
.43 278
.98 345
.44933
.55 067
.01 655
.56 722
17
44
.43 323
.98 342
.44981
.55 019
.01 658
.56 677
16
45
9.43 367
9.98 338
9.45 029
0.54 971
0.01 662
0.56 633
15
46
.43 412
.98 334
.45 078
.54922
.01 666
.56588
14
47
.43 457
.98 331
.45 126
.54874
.01 669
.56 543
13
48
.43 502
.98 327
.45 174
.54826
.01 673
.56 498
12
49
.43546
.98 324
.45 222
.54778
.01 676
.56454
11
50
9.43 591
9.98 320
9.45 271
0.54 729
0.01 680
0.56 409
10
51
.43 635
.98 317
.45 319
.54 681
.01 683
.56 365
9
52
.43 680
.98 313
.45 367
.54 633
.01 687
.56 320
8
53
.43 724
.98 309
.45 415
.54 585
.01 691
.56 276
7
54
.43 769
.98 306
.45 463
.54537
.01 694
.56 231
6
55
9.43 813
9.98 302
9.45511
0.54 489
0.01 698
0.56 187
5
56
.43 857
.98 299
.45 559
.54441
.01 701
.56 143
4
57
.43 901
.98 295
.45606
.54 394
.01 705
.56 099
3
58
.43946
.98 291
.45654
.54346
.01 709
.56 054
2
59
.43 990
.98 288
.45 702
.54298
.01 712
.56 010
1
60
9.44 034
9.98 284
9.45 750
0.54 250
0.01 716
0.55 966
0
Cos
Sin
Cot
Tan
Csc
Sec
'
105° (285°)
(254°) 74°
212
Table 4. Trigonometric Logarithms
16° (196°)
(343°) 163°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.44 034
9.98 284
9.45 750
0.54 250
0.01 716
0.55 966
60
1
.44 078
.98 281
.45 797
.54 203
.01 719
.55 922
59
2
.44 122
.98 277
.45845
.54 155
.01 723
.55 878
58
3
.44 166
.98 273
.45 892
.54 108
.01 727
.55 834
57
4
.44 210
.98 270
.45 940
.54 060
.01 730
.55 790
56
5
9.44 253
9.98 266
9.45 987
0.54 013
0.01 734
0.55 747
55
6
.44 297
.98 262
.46 035
.53 965
.01 738
.55 703
54
7
.44 341
.98 259
.46 082
.53 918
.01 741
.55 659
53
8
.44 385
.98 255
.46 130
.53 870
.01 745
.55 615
52
9
.44 428
.98 251
.46 177*
.53 823
.01 749
.55 572
51
10
9.44 472
9.98 248
9.46 224
0.53 776
0.01 752
0.55 528
50
11
.44516
.98 244
.46 271
.53 729
.01 756
.55484
49
12
.44 559
.98 240
.46 319
.53 681
.01 760
.55 441
48
13
.44 602
.98 237
.46 366
.53 634
.01 763
.55 398
47
14
.44 646
.98 233
.46 413
.53 587
.01 767
.55 354
46
15
9.44 689
9.98 229
9.46 460
0.53 540
0.01 771
0.55311
45
16
.44 733
.98 226
.46 507
.53 493
.01 774
.55 267
44
17
.44 776
.98 222
.46 554
.53 446
.01 778
.55 224
43
18
.44 819
.98 218
.46 601
.53 399
.01 782
.55 181
42
19
.44 862
.98 215
.46 648
.53 352
.01 785
.55 138
41
20
9.44 905
9.98211
9.46 694
0.53 306
0.01 789
0.55 095
40
21
.44 948
.98 207
.46 741
.53 259
.01 793
.55 052
39
22
.44 992
.98 204
.46 788
.53 212
.01 796
.55 008
38
23
.45 035
.98 200
.46 835
.53 165
.01 800
.54965
37
24
.45 077
.98 196
.46 881
.53 119
.01 804
.54 923
36
25
9.45 120
9.98 192
9.46 928
0.53 072
0.01 808
0.54 880
35
26
.45 163
.98 189
.46 975
.53 025
.01811
.54837
34
27
.45 206
.98 185
.47 021
.52 979
.01 815
.54 794
33
28
.45 249
.98 181
.47 068
.52 932
.01 819
.54 751
32
29
.45 292
.98 177
.47 114
.52 886
.01 823
.54 708
31
30
9.45 334
9.98 174
9.47 160
0.52 840
0.01 826
0.54 666
30
31
.45 377
.98 170
.47 207
.52 793
.01 830
.54 623
29
32
.45 419
.98 166
.47 253
.52 747
.01 834
.54 581
28
33
.45 462
.98 162
.47 299
.52 701
.01 838
.54 538
27
34
.45 504
.98 159
.47 346
.52 654
.01 841
.54 496
26
35
9.45 547
9.98 155
9.47 392
0.52 608
0.01 845
0.54 453
25
36
.45 589
.98 151
.47 438
.52 562
.01 849
.54411
24
37
.45 632
.98 147
.47484
.52 516
.01 853
.54 368
23
38
.45 674
.98 144
.47 530
.52 470
.01 856
.54 326
22
39
.45 716
.98 140
.47 576
.52 424
.01 860
.54 284
21
40
9.45 758
9.98 136
9.47 622
0.52 378
0.01 864
0.54 242
20
41
.45 801
.98 132
.47-668
.52 332
.01 868
.54 199
19
42
.45843
.98 129
.47 714
.52 286
.01 871
.54 157
18
43
.45 885
.98 125
.47 760
.52 240
.01 875
.54 115
17
44
.45 927
.98 121
.47 806
.52 194
.01 879
.54 073
16
45
9.45 969
9.98 117
9.47 852
0.52 148
0.01 883
0.54 031
15
46
.46011
.98 113
.47 897
.52 103
.01 887
.53 989
14
47
.46 053
.98 110
.47 943
.52 057
.01 890
.53 947
13
48
.46 095
.98 106
.47 989
.52011
.01 894
.53 905
12
49
.46 136
.98 102
.48 035
.51 965
.01 898
.53 864
11
50
9.46 178
9.98 098
9.48 080
0.51 920
0.01 902
0.53 822
10
51
.46 220
.98 094
.48 126
.51 874
.01 906
.53 780
9
52
.46 262
.98 090
.48 171
.51 829
.01 910
.53 738
8
53
.46 303
.98 087
.48 217
.51 783
.01 913
.53 697
7
54
.46 345
.98 083
.48 262
.51 738
.01 917
.53 655
6
55
9.46 386
9.98079
9.48 307
0.51 693
0.01 921
0.53 614
5
56
.46 428
.98 075
.48 353
.51 647
.01 925
.53 572
4
57
.46 469
.98 07i
.48 398
.51 602
.01 929
.53 531
3
58
.46511
.98 067
.48 443
.51 557
.01 933
.53 489
2
59
.46 552
.98 063
.48 489
.51511
.01 937
.53 448
1
60
9.46 594
9.98 060
9.48 534
0.51 466
0.01 940
0.53 406
0
Cos
Sin
Cot
Tan
Csc
Sec
'
106° (286°)
(253°) 73°
Table 4. Trigonometric Logarithms
213
17° (197°)
(342°) 162C
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.46 594
9.98 060
9.48 534
0.51 466
0.01 940
0.53 406
60
1
.46 635
.98 056
.48 579
.51 421
.01 944
.53 365
59
2
.46 676
.98 052
.48 624
.51 376
.01 948
.53 324
58
3
.46 717
.98 048
.48 669
.51 331
.01 952
.53283
57
4
.46 758
.98044
.48 714
.51 286
.01 956
.53 242
56
5
9.46 800
9.98 040
9.48 759
0.51 241
0.01 960
0.53 200
55
6
.46841
.98 036
.48804
.51 196
.01 964
.53 159
54
7
.46 882
.98 032
.48849
.51 151
.01 968
.53 118
53
8
.46 923
.98 029
.48 894
.51 106
.01 971
.53 077
52
9
.46964
.98 025
.48 939
.51 061
.01 975
.53 036
51
10
9.47 005
9.98 021
9.48 984
0.51 016
0.01 979
0.52 995
50
11
.47 045
.98 017
.49 029
.50 971
.01983
.52 955
49
12
.47 086
.98 013
.49 073
.50 927
.01 987
.52 914
48
13
.47 127
.98 009
.49 118
.50 882
.01 991
.52 873
47
14
.47 168
.98005
.49 163
.50837
.01 995
.52832
46
15
9.47 209
9.98 001
9.49 207
0.50 793
0.01 999
0.52 791
45
16
.47 249
.97 997
.49 252
.50 748
.02 003
.52 751
44
17
.47 290
.97 993
.49 296
.50704
.02 007
.52 710
43
18
.47 330
.97 989
.49 341
.50 659
.02011
.52 670
42
19
.47 371
.97 986
.49 385
.50 615
.02 014
.52 629
41
20
9.47411
9.97 982
9.49 430
0.50 570
0.02 018
0.52 589
40
21
.47 452
.97 978
.49 474
.50 526
.02 022
.52548
39
22
.47 492
.97 974
.49 519
.50 481
.02 026
.52 508
38
23
.47 533
.97 970
.49 563
.50 437
.02 030
.52 467
37
24
.47 573
.97 966
.49 607
.50 393
.02 034
.52 427
36
25
9.47 613
9.97 962
9.49 652
0.50 348
0.02 038
0.52 387
35
26
.47 654
.97 958
.49 696
.50304
.02042
.52 346
34
27
.47 694
.97 954
.49 740
.50 260
.02 046
.52 306
33
28
.47 734
.97 950
.49784
.50 216
.02 050
.52 266
32
29
.47 774
.97 946
.49 828
.50 172
.02 054
.52 226
31
30
9.47 814
9.97 942
9.49 872
0.50 128
0.02 058
0.52 186
30
31
.47 854
.97 938
.49 916
.50084
.02 062
.52 146
29
32
.47 894
.97 934
.49 960
.50040
.02 066
.52 106
28
33
.47 934
.97 930
.50 004
.49 996
.02 070
.52 066
27
34
.47 974
.97 926
.50 048
.49 952
.02 074
.52 026
26
35
9.48 014
9.97 922
9.50 092
0.49 908
0.02 078
0.51 986
25
30
.48 054
.97 918
.50 136
.49864
.02 082
.51 946
24
37
.48 094
.97 914
.50 180
.49 820
.02 086
.51 906
23
38
.48 133
.97 910
.50 223
.49 777
.02 090
.51 867
22
39
.48 173
.97 906
.50 267
.49 733
.02 094
.51 827
21
40
9.48 213
9.97 902
9.50311
0.49 689
0.02 098
0.51 787
20
41
.48 252
.97 898
.50 355
.49 645
.02 102
.51 748
19
42
.48 292
.97 894
.50 398
.49 602
.02 106
.51 708
18
43
.48 332
.97 890
.50442
.49 558
.02 110
.51 668
17
44
.48 371
.97 886
.50 485
.49 515
.02 114
.51 629
16
45
9.48411
9.97 882
9.50 529
0.49 471
0.02 118
0.51 589
15
46
.48 450
.97 878
.50 572
.49 428
.02 122
.51 550
14
47
.48 490
.97 874
.50 616
.49384
.02 126
.51 510
13
48
.48 529
.97 870
.50 659
.49 341
.02 130
.51 471
12
49
.48 568
.97 866
.50 703
.49 297
.02 134
.51 432
11
50
9.48 607
9.97 861
9.50 746
0.49 254
0.02 139
0.51 393
10
51
.48 647
.97 857
.50 789
.49211
.02 143
.51 353
9
52
.48 686
.97853
.50833
.49 167
.02 147
.51 314
8
53
.48 725
.97849
.50 876
.49 124
.02 151
.51 275
7
54
.48764
.97845
.50 919
.49 081
.02 155
.51 236
6
55
9.48 803
9.97 841
9.50 962
0.49 038
0.02 159
0.51 197
5
56
.48842
.97 837
.51 005
.48 995
.02 163
.51 158
4
57
.48 881
.97 833
.51048
.48 952
.02 167
.51 119
3
58
.48 920
.97 829
.51 092
.48 908
.02 171
.51080
2
59
.48 959
.97 825
.51 135
.48 865
.02 175
.51041
1
60
9.48 998
9.97 821
9.51 178
0.48 822
0.02 179
0.51 002
0
Cos
Sin
Cot
Tan
Csc
Sec
'
107° (287°)
(252°) 72°
214
Table 4. Trigonometric Logarithms
18° (198°)
(341°) 161°
'
Sin
Cos
Tan
Cot
See
Csc
0
9.48 998
9.97 821
9.51 178
0.48 822
0.02 179
0.51 002
60
1
.49 037
.97 817
.51 221
.48 779
.02 183
.50 963
59
2
.49 076
.97 812
.51264
.48 736
.02 188
.50 924
58
3
.49 115
.97 808
.51 306
.48 694
.02 192
.50 885
57
4
.49 153
.97804
.51 349
.48651
.02 196
.50847
56
5
9.49 192
9.97 800
9.51 392
0.48 608
0.02 200
0.50 808
55
6
.49 231
.97 796
.51 435
.48 565
.02204
.50 769
54
7
.49 269
.97 792
.51 478
.48 522
.02 208
.50 731
53
8
.49 308
.97 788
.51 520
.48 480
.02 212
.50 692
52
9
.49 347
.97784
.51 563
.48 437
.02 216
.50 653
51
10
9.49 385
9.97 779
9.51 606
0.48 394
0.02 221
0.50 615
50
11
.49 424
.97 775
.51648
.48 352
.02 225
.50 576
49
12
.49 462
.97 771
.51 691
.48 309
.02 229
.50 538
48
13
.49 500
.97 767
.51 734
.48 266
.02 233
.50 500
47
14
.49 539
.97 763
.51 776
.48 224
.02 237
.50 461
46
15
9.49 577
9.97 759
9.51 819
0.48 181
0.02 241
0.50 423
45
16
.49 615
.97 754
.51 861
.48 139
.02 246
.50 385
44
17
.49 654
.97 750
.51 903
.48 097
.02 250
.50 346
43
18
.49 692
.97 746
.51 946
.48 054
.02 254
.50 308
42
19
.49 730
.97 742
.51 988
.48 012
.02 258
.50 270
41
20
9.49 768
9.97 738
9.52 031
0.47 969
0.02 262
0.50 232
40
21
.49 806
.97 734
.52 073
.47 927
.02 266
.50 194
39
22
.49844
.97 729
.52 115
.47885
.02 271
.50 156
38
23
.49 882
.97 725
.52 157
.47843
.02 275
.50 118
37
24
.49 920
.97 721
.52200
.47800
.02 279
.50 080
36
25
9.49 958
9.97 717
9.52 242
0.47 758
0.02 283
0.50 042
35
26
.49 996
.97 713
.52284
.47 716
.02 287
.50 004
34
27
.50 034
.97 708
.52 326
.47 674
.02 292
.49 966
33
28
.50 072
.97704
.52 368
.47 632
.02 296
.49 928.
32
29
.50 110
.97 700
.52 410
.47 590
.02 300
.49 890
31
30
9.50 148
9.97 696
9.52 452
0.47 548
0.02 304
0.49 852
30
31
.50 185
.97 691
.52 494
.47 506
.02 309
.49 815
29
32
.50 223
.97 687
.52 536
.47464
.02 313
.49 777
28
33
.50 261
.97 683
.52 578
.47 422
.02 317
.49 739
27
34
.50 298
.97 679
.52 620
.47 380
.02 321
.49 702
26
35
9.50 336
9.97 674
9.52 661
0.47 339
0.02 326
0.49 664
25
36
.50 374
.97 670
.52 703
.47 297
.02 330
.49 626
24
37
.50411
.97 666
.52 745
.47 255
.02 334
.49 589
23
38
.50 449
.97 662
.52 787
.47 213
.02 338
.49 551
22
39
.50 486
.97 657
.52 829
.47 171
.02 343
.49 514
21
40
9.50 523
9.97 653
9.52 870
0.47 130
0.02 347
0.49 477
20
41
.50 561
.97 649
.52 912
.47 088
.02 351
.49 439
19
42
.50 598
.97645
.52 953
.47047
.02 355
.49 402
18
43
.50 635
.97640
.52 995
.47 005
.02 360
.49 365
17
44
.50 673
.97 636
.53 037
.46 963
.02364
.49 327
16
45
9.50 710
9.97 632
9.53 078
0.46 922
0.02 368
0.49 290
15
46
.50 747
.97 628
.53 120
.46 880
.02 372
.49 253
14
47
.50784
.97 623
.53 161
.46839
.02 377
.49 216
13
48
.50 821
.97 619
.53 202
.46 798
.02 381
.49 179
12
49
.50 858
.97 615
.53 244
.46 756
.02 385
.49 142
11
50
9.50 896
9.97 610
9.53 285
0.46 715
0.02 390
0.49 104
10
51
.50 933
.97 606
.53 327
.46 673
.02 394
.49 067
9
52
.50 970
.97 602
.53 368
.46 632
.02 398
.49 030
8
53
.51 007
.97 597
.53 409
.46 591
.02 403
.48 993
7
54
.51 043
.97 593
.53 450
.46 550
.02 407
.48 957
6
55
9.51 080
9.97 589
9.53 492
0.46 508
0.02 411
0.48 920
5
56
.51 117
.97584
.53 533
.46 467
.02 416
.48 883
4
57
.51 154
.97 580
.53 574
.46 426
.02 420
.48846
3
58
.51 191
.97 576
.53 615
.46 385
.02 424
.48 809
2
59
.51 227
.97 571
.53 656
.46344
.02 429
.48 773
1
60
9.51 264
9.97 567
9.53 697
0.46 303
0.02 433
0.48 736
0
Cos
Sin
Cot
Tan
Csc
Sec
'
108° (288°)
(251°) 71°
Table 4. Trigonometric Logarithms
215
19° (199°)
(340°) 160°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.51 264
9.97 567
9.53 697
0.46 303
0.02 433
0.48 736
60
1
.51 301
.97 563
.53 738
.46 262
.02 437
.48 699
59
2
.51 338
.97 558
.53 779
.46 221
.02442
.48 662
58
3
.51 374
.97 554
.53 820
.46 180
.02 446
.48 626
57
4
.51 411
.97 550
.53 861
.46 139
.02 450
.48 589
56
5
9.51 447
9.97 545
9.53 902
0.46 098
0.02 455
0.48 553
55
6
.51484
.97 541
.53 943
.46 057
.02 459
.48 516
54
7
.51 520
.97 536
.53984
.46 016
.02 464
.48 480
53
8
.51 557
.97 532
.54 025
.45 975
.02 468
.48443
52
9
.51 593
.97 528
.54065
.45935
.02 472
.48 407
51
10
9.51 629
9.97 523
9.54 106
0.45 894
0.02 477
0.48 371
50
11
.51 666
.97 519
.54147
.45853
.02 481
.48 334
49
12
.51 702
.97 515
.54 187
.45 813
.02 485
.48 298
48
13
.51 738
.97 510
.54 228
.45 772
.02 490
.48 262
47
14
.51 774
.97 506
.54 269
.45 731
.02 494
.48 226
46
15
9.51 811
9.97 501
9.54 309
0.45 691
0.02 499
0.48 189
45
16
.51 847
.97 497
.54350
.45 650
.02 503
.48 153
44
17
.51883
.97 492
.54390
.45 610
.02 508
.48 117
43
18
.51 919
.97488
.54431
.45 569
.02 512
.48 081
42
19
.51 955
.97484
.54 471
.45 529
.02 516
.48 045
41
20
9.51 991
9.97 479
9.54 512
0.45 488
0.02 521
0.48 009
40
21
.52 027
.97 475
.54 552
.45448
.02 525
.47 973
39
22
.52 063
.97 470
.54593
.45 407
.02 530
.47 937
38
23
.52 099
.97 466
.54 633
.45 367
.02 534
.47 901
37
24
.52 135
.97 461
.54673
.45 327
.02 539
.47 865
36
25
9.52 171
9.97457
9.54 714
0.45 286
0.02 543
0.47 829
35
26
.52 207
.97 453
.54754
.45246.
.02 547
.47 793
34
27
.52 242
.97448
.54 794
.45206
.02 552
.47 758
33
28
.52 278
.97444
.54835
.45 165
.02 556
.47 722
32
29
.52 314
.97 439
.54 875
.45 125
.02 561
.47 686
31
30
9.52 350
9.97 435
9.54 915
0.45 085
0.02 565
0.47 650
30
31
.52 385
.97 430
.54 955
.45045
.02 570
.47 615
29
32
.52421
.97 426
.54995
.45 005
.02 574
.47 579
28
33
.52 456
.97421
.55 035
.44965
.02 579
.47 544
27
34
.52 492
.97 417
.55 075
.44 925
.02 583
.47 508
26
35
9.52 527
9.97 412
9.55 115
0.44 885
0.02 588
0.47 473
25
36
.52 563
.97 408
.55 155
.44845
.02 592
.47 437
24
37
.52 598
.97 403
.55 195
.44805
.02 597
.47 402
23
38
.52 634
.97 399
.55 235
.44765
.02 601
.47 366
22
39
.52 669
.97 394
.55 275
.44725
.02 606
.47 331
21
40
9.52 705
9.97 390
9.55 315
0.44685
0.02 610
0.47 295
20
41
.52 740
.97 385
.55 355
.44645
.02 615
.47 260
19
42
.52 775
.97 381
.55 395
.44605
.02 619
.47 225
18
43
.52811
.97 376
.55 434
.44566
.02 624
.47 189
17
44
.52846
.97 372
.55 474
.44526
.02 628
.47 154
16
45
9.52 881
9.97 367
9.55 514
0.44 486
0.02 633
0.47 119
15
46
.52 916
.97 363
.55554
.44 446
.02 637
.47084
14
47
.52 951
.97 358
.55 593
.44407
.02642
.47049
13
48
.52 986
.97 353
.55 633
.44367
.02647
.47 014
12
49
.53 021
.97 349
.55 673
.44327
.02 651
.46 979
11
50
9.53 056
9.97 344
9.55 712
0.44 288
0.02 656
0.46 944
10
51
.53 092
.97 340
.55 752
.44248
.02 660
.46 908
9
52
.53 126
.97 335
.55 791
.44209
.02 665
.46 874
8
53
.53 161
.97 331
.55831
.44169
.02 669
.46 839
7
54
.53 196
.97 326
.55 870
.44 130
.02 674
.46804
6
55
9.53 231
9.97 322
9.55 910
0.44 090
0.02 678
0.46 769
5
56
.53 266
.97 317
.55 949
.44051
.02 683
.46 734
4
57
.53 301
.97 312
.55 989
.44011
.02688
.46 699
3
58
.53 336
.97 308
.56 028
.43 972
.02 692
.46664
2
59
.53 370
.97 303
.56 067
.43 933
.02 697
.46 630
1
60
9.53 405
9.97 299
9.56 107
9.43 893
0.02 701
0.46 595
0
Cos
Sin
Cot
Tan
Csc
Sec
'
109° (289°)
(250°) 70°
216
Table 4. Trigonometric Logarithms
20° (200°)
(339°) 159°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.53 405
9.97 299
9.56 107
0.43 893
0.02 701
0.46 595
60
1
.53 440
.97 294
.56 146
.43 854
.02 706
.46 560
59
2
.53 475
.97 289
.56 185
.43 815
.02711
.46 525
58
3
.53 509
.97 285
.56 224
.43 776
.02 715
.46 491
57
4
.53 544
.97 280
.56 264
.43 736
.02 720
.46 456
56
5
9.53 578
9.97 276
9.56 303
0.43 697
0.02 724
0.46 422
55
6
.53 613
.97 271
.56 342
.43 658
.02 729
.46 387
54
7
.53 647
.97 266
.56 381
.43 619
.02 734
.46 353
53
8
.53 682
.97 262
.56 420
.43 580
.02 738
.46 318
52
9
.53 716
.97 257
.56 459
.43 541
.02 743
.46284
51
10
9.53 751
9.97 252
9.56 498
0.43 502
0.02 748
0.46 249
50
11
.53 785
.97 248
.56 537
.43 463
.02 752
.46 215
49
12
.53 819
.97 243
.56 576
.43 424
.02 757
.46 181
48
13
.53 854
.97 238
.56 615
.43 385
.02 762
.46 146
47
14
.53 888
.97 234
.56 654
.43 346
.02 766
.46 112
46
15
9.53 922
9.97 229
9.56 693
0.43 307
0.02 771
0.46 078
45
16
.53 957
.97 224
.56 732
.43 268
.02 776
.46 043
44
17
.53 991
.97 220
.56 771
.43 229
.02 780
.46 009
43
18
.54 025
.97 215
.56 810
.43 190
.02 785
.45 975
42
19
.54 059
.97 210
.56849
.43 151
.02 790
.45 941
41
20
9.54 093
9.97 206
9.56 887
0.43 113
0.02 794
0.45 907
40
21
.54 127
.97 201
.56 926
.43 074
.02 799
.45 873
39
22
.54 161
.97 196
.56 965
.43 035
.02 804
.45 839
38
23
.54 195
.97 192
.57 004
.42 996
.02 808
.45 805
37
24
.54 229
.97 187
.57 042
.42 958
.02 813
.45 771
36
25
9.54 263
9.97 182
9.57 081
0.42 919
0.02 818
0.45 737
35
26
.54 297
.97 178 .
.57 120
.42 880
.02 822
.45 703
34
27
.54 331
.97 173
.57 158
.42842
.02 827
.45 669
33
28
.54 365
.97 168
.57 197
.42 803
.02 832
.45 635
32
29
.54 399
.97 163
.57 235
.42 765
.02 837
.45 601
31
30
9.54 433
9.97 159
9.57 274
0.42 726
0.02 841
0.45 567
30
31
.54466
.97 154
.57 312
.42 688
.02846
.45 534
29
32
.54 500
.97 149
.57 351
.42649
.02 851
.45 500
28
33
.54 534
.97 145
.57 389
.42611
.02 855
.45 466
27
34
.54 567
.97 140
.57 428
.42 572
.02 860
.45 433
26
35
9.54 601
9.97 135
9.57 466
0.42 534
0.02 865
0.45 399
25
36
.54 635
.97 130
.57504
.42 496
.02 870
.45 365
24
37
.54 668
.97 126
.57 543
.42 457
.02 874
.45 332
23
38
.54 702
.97 121
.57 581
.42 419
.02 879
.45 298
22
39
.54 735
.97 116
.57 619
.42 381
.02 884
.45 265
21
40
9.54 769
9.97 111
9.57 658
0.42 342
0.02 889
0.45 231
20
41
.54 802
.97 107
.57 696
.42 304
.02 893
.45 198
19 .
42
.54 836
.97 102
.57 734
.42 266
.02 898
.45 164
18
43
.54869
.97 097
.57 772
.42 228
.02 903
.45 131
17
44
.54 903
.97 092
.57 810
.42 190
.02 908
.45 097
16
45
9.54 936
9.97 087
9.57 849
0.42 151
0.02 913
0.45 064
15
46
.54 969
.97 083
.57 887
.42 113
.02 917
.45 031
14
47
.55 003
.97 078
.57 925
.42 075
.02 922
.44 997
13
48
.55 036
.97 073
.57 963
.42 037
.02 927
.44 964
12
49
.55 069
.97 068
.58 001
.41 999
.02 932
.44 931
11
50
9.55 102
9.97 063
9.58 039
0.41 961
0.02 937
0.44 898
10
51
.55 136
.97 059
.58 077
.41 923
.02 941
.44864
9
52
.55 169
.97 054
.58 115
.41 885
.02 946
.44 831
8
53
.55 202
.97 049
.58 153
.41 847
.02 951
.44 798
7
54
.55 235
.97 044
.58 191
.41 809
.02 956
.44 765
6
55
9.55 268
9.97 039
9.58 229
0.41 771
0.02 961
0.44 732
5
56
.55 301
.97 035
.58 267
.41 733
.02 965
.44 699
4
57
.55 334
.97 030
.58 304
.41 696
.02 970
.44 666
3
58
.55 367
.97 025
.58 342
.41 658
.02 975
.44633
2
59
.55 400
.97 020
.58 380
.41 620
.02 980
.44 600
1
60
9.55 433
9.97 015
9.58418
0.41 582
0.02 985
0.44 567
0
Cos
Sin
Cot
Tan
Csc
Sec
'
110° (290°)
(249°) 69°
Table 4. Trigonometric Logarithms
217
21° (201°)
(338°) 158°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.55 433
9.97 015
9.58418
0.41 582
0.02 985
0.44 567
60
1
.55 466
.97 010
.58 455
.41 545
.02990
.44534
59
2
.55 499
.97 005
.58 493
.41 507
.02 995
.44501
58
3
.55 532
.97 001
.58 531
.41 469
.02 999
.44468
57
4
.55 564
.96 996
.58 569
.41 431
.03 004
.44436
56
5
9.55 597
9.96 991
9.58 606
0.41 394
0.03 009
0.44 403
55
6
.55 630
.96 986
.58 644
.41 356
.03 014
.44370
54
7
.55 663
.96 981
.58 681
.41 319
.03 019
.44 337
53
8
.55 695
.96 976
.58 719
.41 281
.03 024
.44 305
52
9
.55 728
.96 971
.58 757
.41 243
.03 029
.44272
51
10
9.55 761
9.96 966
9.58 794
0.41 206
0.03 034
0.44 239
50
11
.55 793
.96 962
.58 832
.41 168
.03 038
.44207
49
12
.55 826
.96 957
.58 869
.41 131
.03 043
.44 174
48
13
.55 858
.96 952
.58 907
.41 093
.03 048
.44 142
47
14
.55 891
.96 947
.58944
.41 056
.03 053
.44109
46
15
9.55 923
9.96 942
9.58 981
0.41 019
0.03 058
0.44 077
45
16
.55 956
.96 937
.59 019
.40 981
.03 063
.44044
44
17
.55 988
.96 932
.59 056
.40 944
.03 068
.44 012
43
18
.56 021
.96 927
.59 094
.40 906
.03 073
.43 979
42
19
.56 053
.96 922
.59 131
.40 869
.03 078
' .43947
41
20
9.56 085
9.96 917
9.59 168
0.40 832
0.03 083
0.43 915
40
21
.56 118
.96 912
.59 205
.40 795
.03 088
.43 882
39
22
.56 150
.96 907
.59 243
.40 757
.03 093
.43 850
38
23
.56 182
.96 903
.59 280
.40 720
.03 097
.43 818
37
24
.56 215
.96 898
.59 317
.40 683
.03 102
.43 785
36
25
9.56 247
9.96 893
9.59 354
0.40 646
0.03 107
0.43 753
35
26
.56 279
.96 888
.59 391
.40 609
.03 112
.43 721
34
27
.56311
.96 883
.59 429
.40 571
.03 117
.43 689
33
28
.56 343
.96 878
.59 466
.40 534
.03 122
.43 657
32
29
.56 375
.96 873
.59 503
.40 497
.03 127
.43 625
31
30
9.56 408
9.96 868
9.59 540
0.40 460
0.03 132
0.43 592
30
31
.56440
.96 863
.59 577
.40 423
.03 137
.43 560
29
32
.56 472
.96 858
.59 614
.40 386
.03 142
.43 528
28
33
.56 504
.96853
.59 651
.40 349
.03 147
.43 496
27
34
.56 536
.96848
.59 688
.40 312
.03 152
.43 464
26
35
9.56 568
9.96 843
9.59 725
0.40 275
0.03 157
0.43 432
25
36
.56 599
.96 838
.59 762
.40 238
.03 162
.43 401
24
37
.56 631
.96833
.59 799
.40 201
.03 167
.43 369
23
38
.56 663
.96 828
.59 835
.40 165
.03 172
.43 337
22
39
.56 695
.96 823
.59 872
.40 128
.03 177
.43 305
21
40
9.56 727
9.96 818
9.59 909
0.40 091
0.03 182
0.43 273
20
41
.56 759
.96 813
.59 946
.40 054
.03 187
.43 241
19
42
.56 790
.96 808
.59 983
.40 017
.03 192
.43 210
18
43
.56 822
.96 803
.60 019
.39 981
.03 197
.43 178
17
44
.56 854
.96 798
.60 056
.39944
.03 202
.43 146
16
45
9.56 886
9.96 793
9.60 093
0.39 907
0.03 207
0.43 114
15
46
.56 917
.96 788
.60 130
.39 870
.03 212
.43 083
14
47
.56 949
.96 783
.60 166
.39 834
.03 217
.43 051
13
48
.56 980
.96 778
.60 203
.39 797
.03 222
.43 020
12
49
.57 012
.96 772
.60 240
.39 760
.03 228
.42 988
11
50
9.57 044
9.96 767
9.60 276
0.39 724
0.03 233
0.42 956
10
51
.57 075
.96 762
.60 313
.39 687
.03 238
.42 925
9
52
.57 107
.96 757
.60 349
.39 651
.03 243
.42 893
8
53
.57 138
.96 752
.60 386
.39 614
.03 248
.42 862
7
54
.57 169
.96 747
.60 422
.39 578
.03 253
.42831
6
55
9.57 201
9.96 742
9.60 459
0.39 541
0.03 258
0.42 799
5
56
.57 232
.96 737
.60 495
.39 505
.03 263
.42 768
4
57
.57 264
.96 732
.60 532
.39 468
.03 268
.42 736
3
58
.57 295
.96 727
.60 568
.39 432
.03 273
.42 705
2
59
.57 326
.96 722
.60 605
.39 395
.03 278
.42 674
1
60
9.57 358
9.96 717
9.60 641
0.39 359
0.03 283
0.42 642
0
Cos
Sin
Cot
Tan
Csc
Sec
'
111° (291°)
(248°) 68°
218
Table 4. Trigonometric Logarithms
22° (202°)
(337°) 157°
/
Sin
Cos
Tan
Cot
Sec
Csc
0
9.57 358
9.96 717
9.60 641
0.39 359
0.03 283
0.42 642
60
1
.57 389
.96 711
.60 677
.39 323
.03 289
.42611
59
2
.57 420
.96 706
.60 714
.39 286
.03 294
.42 580
58
3
.57 451
.96 701
.60 750
.39 250
.03 299
.42 549
57
4
.57 482
.96 696
.60 786
.39 214
.03304
.42 518
56
5
9.57 514
9.96 691
9.60 823
0.39 177
0.03 309
0.42 486
55
6
.57 545
.96 686
.60859
.39 141
.03 314
.42 455
54
7
.57 576
.96 681
.60 895
.39 105
.03 319
.42 424
53
8
.57 607
.96 676
.60 931
.39 069
.03 324
.42 393
52
9
.57 638
.96 670
.60 967
.39 033
.03 330
.42 362
51
10
9.57 669
9.96 665
9.61 004
9.38 996
0.03 335
0.42 331
50
11
.57 700
.96 660
.61040
.38 960
.03 340
.42 300
49
12
.57 731
.96 655
.61 076
.38 924
.03 345
.42 269
48
13
.57 762
.96 650
.61 112
.38 888
.03 350
.42 238
47
14
.57 793
.96645
.61 148
.38 852
.03 355
.42 207
46
15
9.57 824
9.96 640
9.61 184
0.38 816
0.03 360
0.42 176
45
16
.57 855
.96 634
.61 220
.38 780
.03 366
.42 145
44
17
.57 885
.96 629
.61 256
.38744
.03 371
.42 115
43
18
.57 916
.96 624
.61 292
.38 708
.03 376
.42084
42
19
.57 947
.96 619
.61 328
.38 672
.03 381
.42 053
41
20
9.57 978
9.96 614
9.61 364
0.38 636
0.03 386
0.42 022
40
21
.58 008
.96 608
.61 400
.38 600
.03 392
.41 992
39
22
.58 039
.96 603
.61 436
.38564
.03 397
.41 961
38
23
.58 070
.96 598
.61 472
.38 528
.03 402
.41 930
37
24
.58 101
.96 593
.61 508
.38 492
.03 407
.41 899
36
25
9.58 131
9.96 588
9.61 544
0.38 456
0.03 412
0.41 869
35
26
.58 162
.96 582
.61 579
.38 421
.03 418
.41 838
34
27
.58 192
.96 577
.61 615
.38 385
.03 423
.41 808
33
28
.58 223
.96 572
.61 651
.38 349
.03 428
.41 777
32
29
.58 253
.96 567
.61 687
.38313
.03 433
.41 747
31
30
9.58 284
9.96 562
9.61 722
0.38 278
0.03 438
0.41 716
30
31
.58 314
.96 556
.61 758
.38 242
.03444
.41 686
29
32
.58 345
.96 551
.61 794
.38 206
.03449
.41 655
28
33
.58 375
.96 546
.61830
.38 170
.03454
.41 625
27
34
.58 406
.96 541
.61 865
.38 135
.03 459
.41 594
26
35
9.58 436
9.96 535
9.61 901
0.38 099
0.03 465
0.41 564
25
36
.58 467
.96 530
.61 936
.38064
.03 470
.41 533
24
37
.58 497
.96 525
.61 972
.38 028
.03 475
.41 503
23
38
.58 527
.96 520
.62 008
.37 992
.03 480
.41 473
22
39
.58 557
.96 514
.62043
.37 957
.03 486
.41 443
21
40
9.58 588
9.96 509
9.62 079
0.37 921
0.03 491
0.41 412
20
41
.58 618
.96 504
.62114
.37 886
.03 496
.41 382
19
42
.58648
.96 498
.62 150
.37 850
.03 502
.41 352
18
43
.58 678
.96 493
.62 185
.37 815
.03 507
.41 322
17
44
.58 709
.96 488
.62 221
.37 779
.03 512
.41 291
16
45
9.58 739
9.96 483
9.62 256
0.37 744
0.03 517
0.41 261
15
46
.58 769
.96 477
.62 292
.37 708
.03 523
.41 231
14
47
.58 799
.96 472
.62 327
.37 673
.03 528
Al 201
13
48
.58 829
.96 467
.62 362
.37 638
.03 533
.41 171
12
49
.58 859
.96 461
.62 398
.37 602
.03 539
.41 141
11
50
9.58 889
9.96 456
9.62 433
0.37 567
0.03 544
0.41 111
10
51
.58 919
.96 451
.62 468
.37 532
.03 549
.41 081
9
52
.58 949
.96445
.62504
.37 496
.03 555
.41 051
8
53
.58 979
.96440
.62 539
.37 461
.03 560
.41 021
7
54
.59 009
.96 435
.62 574
.37 426
.03 565
.40 991
6
55
9.59 039
9.96 429
9.62 609
0.37 391
0.03 571
0.40 961
5
56
.59 069
.96 424
.62 645
.37 355
.03 576
.40 931
4
57
.59 098
.96 419
.62 680
.37 320
.03 581
.40 902
3
58
.59 128
.96 413
.62 715
.37 285
.03 587
.40 872
2
59
.59 158
.96 408
.62 750
.37 250
.03 592
.40842
1
60
9.59 188
9.96 403
9.62 785
0.37 215
0.03 597
0.40 812
0
Cos
Sin
Cot
Tan
Csc
Sec
'
112 r (292°)
(247°) 67°
Table 4. Trigonometric Logarithms
219
83° (203°)
(336°) 156°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.59 188
9.96 403
9.62 785
0.37 215
0.03 597
0.40 812
60
1
.59 218
.96 397
.62 820
.37 180
.03 603
.40 782
59
2
.59 247
.96 392
.62855
.37 145
.03 608
.40 753
58
3
.59 277
.96 387
.62 890
.37 110
.03 613
.40 723
57
4
.59 307
.96 381
.62 926
.37 074
.03 619
.40 693
56
5
9.59 336
9.96 376
9.62 961
0.37 039
0.03 624
0.40664
55
6
.59 366
.96 370
.62 996
.37004
.03 630
.40 634
54
7
.59 396
.96 365
.63 031
.36 969
.03 635
.40604
53
8
.59 425
.96 360
.63 066
.36 934
.03 640
.40 575
52
9
.59 455
.96 354
.63 101
.36 899
.03 646
.40545
51
10
9.59 484
9.96 349
9.63 135
0.36 865
0.03 651
0.40 516
50
11
.59 514
.96 343
.63 170
.36830
.03 657
.40 486
49
12
' .59 543
.96 338
.63 205
.36 795
.03 662
.40 457
48
13
.59 573
.96 333
.63 240
.36 760
.03 667
.40 427
47
14
.59 602
.96 327
.63 275
.36 725
.03 673
.40 398
46
15
9.59 632
9.96 322
9.63 310
0.36 690
0.03 678
0.40 368
45
16
.59 661
.96 316
.63 345
.36 655
.03684
.40 339
44
17
.59 690
.96311
.63 379
.36 621
.03 689
.40 310
43
18
.59 720
.96 305
.63 414
.36 586
.03 695
.40 280
42
19
.59 749
.96 300
.63 449
.36 551
.03 700
.40 251
41
20
9.59 778
9.96 294
9.63 484
0.36 516
0.03 706
0.40 222
40
21
.59 808
.96 289
.63 519
.36 481
.03711
.40 192
39
22
.59 837
.96284
.63553
.36 447
.03 716
.40 163
38
23
.59 866
.96 278
.63 588
.36412
.03 722
.40 134
37
24
.59 895
.96 273
.63 623
.36 377
.03 727
.40 105
36
25
9.59 924
9.96 267
9.63 657
0.36 343
0.03 733
0.40 076
35
26
.59 954
.96 262
.63 692
.36 308
.03 738
.40 046
34
27
.59983
.96 256
.63 726
.36 274
.03 744
.40 017
33
28
.60 012
.96 251
.63 761
.36 239
.03 749
.39 988
32
29
.60 041
.96 245
.63 796
.36 204
.03 755
.39 959
31
30
9.60 070
9.96 240
9.63 830
0.36 170
0.03 760
.39 930
30
31
.60 099
.96 234
.63 865
.36 135
.03 766
.39 901
29
32
.60 128
.96 229
.63 899
.36 101
.03 771
.39 872
28
33
.60 157
.96 223
.63 934
.36 066
.03 777
.39843
27
34
.60 186
.96 218
.63 968
.36 032
.03 782
.39 814
26
35
9.60 215
9.96 212
9.64 003
0.35 997
0.03 788
0.39 785
25
36
.60 244
.96 207
.64 037
.35 963
.03 793
.39 756
24
37
.60 273
.96 201
.64072
.35 928
.03 799
.39 727
23
38
.60 302
.96 196
.64 106
.35 894
.03804
.39 698
22
39
.60 331
.96 190
.64 140
.35 860
.03 810
.39 669
21
40
9.60 359
9.96 185
9.64 175
0.35 825
0.03 815
0.39 641
20
41
.60 388
.96 179
.64209
.35 791
.03 821
.39 612
19
42
.60 417
.96 174
.64243
.35 757
.03 826
.39 583
18
43
.60446
.96 168
.64278
.35 722
.03832
.39 554
17
44
.60 474
.96 162
.64 312
.35688
.03838
.39 526
16
45
9.60 503
9.96 157
9.64 346
0.35 654
0.03 843
0.39 497
15
46
.60 532
.96 151
.64381
.35 619
.03849
.39 468
14
47
.60 561
.96 146
.64415
.35 585
.03854
.39 439
13
48
.60589
.96 140
.64449
.35 551
.03 860
.39411
12
49
.60 618
.96 135
.64483
.35 517
.03 865
.39 382
11
50
9.60 646
9.96 129
9.64517
0.35 483
0.03 871
0.39 354
10
51
.60 675
.96 123
.64552
.35448
.03 877
.39 325
9
52
.60704
.96 118
.64586
.35 414
.03 882
.39 296
8
53
.60 732
.96112
.64620
.35380
.03888
.39 268
7
54
.60 761
.96 107
.64654
.35 346
.03 893
.39 239
6
55
9.60 789
9.96 101
9.64688
0.35 312
0.03 899
0.39211
5
56
.60 818
.96095
.64722
.35 278
.03 905
.39 182
4
57
.60846
.96090
.64756
.35244
.03 910
.39 154
3
58
.60 875
.96084
.64790
.35 210
.03 916
.39 125
2
59
.60 903
.96 079
.64824
.35 176
.03 921
.39097
1
60
9.60 931
9.96 073
9.64 858
0.35 142
0.03 927
0.39 069
0
Cos
Sin
Cot
Tan
Csc
Sec
'
113° (293°)
(246°) 66°
220
Table 4. Trigonometric Logarithms
24° (204°)
(335°) 155°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.60 931
9.96 073
9.64 858
0.35 142
0.03 927
0.39 069
60
1
.60 960
.96 067
.64 892
.35 108
.03 933
.39 040
59
2
.60 988
.96 062
.64 926
.35 074
.03 938
.39 012
58
3
.61 016
.96 056
.64960
.35 040
.03 944
.38 984
57
4
.61045
.96 050
.64994
.35 006
.03 950
.38 955
56
5
9.61 073
9.96 045
9.65 028
0.34 972
0.03 955
0.38 927
55
6
.61 101
.96 039
.65 062
.34 938
.03 961
.38 899
54
7
.61 129
.96 034
.65 096
.34 904
.03 966
.38 871
53
8
.61 158
.96 028
.65 130
.34 870
.03 972
.38 842
52
9
.61 186
.96 022
.65 164
.34 836
.03 978
.38 814
51
10
9.61 214
9.96 017
9.65 197
0.34 803
0.03 983
0.38 786
50
11
.61 242
.96011
.65 231
.34 769
.03 989
.38 758
49
12
.61 270
.96 005
.65 265
.34 735
.03 995
.38 730
48
13
.61 298
.96 000
.65 299
.34 701
.04000
.38 702
47
14
.61 326
.95 994
.65 333
.34 667
.04006
.38 674
46
15
9.61 354
9.95 988
9.65 366
0.34 634
0.04 012
0.38 646
45
16
.61 382
.95 982
.65 400
.34600
.04018
.38 618
44
17
.61411
.95 977
.65 434
.34 566
.04 023
.38 589
43
18
.61 438
.95 971
.65 467
.34 533
.04029
.38 562
42
19
.61 466
.95 965
.65 501
.34 499
.04035
.38 534
41
20
9.61 494
9.95 960
9.65 535
0.34 465
0.04 040
0.38 506
40
21
.61 522
.95 954
.65 568
.34 432
.04 046
.38 478
39
22
.61 550
.95 948
.65 602
.34398
.04052
.38 450
38
23
.61 578
.95 942
.65 636
.34 364
.04058
.38 422
37
24
.61 606
.95 937
.65 669
.34 331
.04 063
.38 394
36
25
9.61 634
9.95 931
9.65 703
0.34 297
0.04 069
0.38 366
35
26
.61 662
.95 925
.65 736
.34 264
.04075
.38 338
34
27
.61 689
.95 920
.65 770
.34 230
.04 080
.38311
33
28
.61 717
.95 914
.65 803
.34 197
.04 086
.38 283
32
29
.61 745
.95 908
.65 837
.34163
.04 092
.38 255
31
30
9.61 773
9.95 902
9.65 870
0.34 130
0.04 098
0.38 227
30
31
.61 800
.95 897
.65 904
.34 096
.04 103
.38 200
29
32
.61 828
.95 891
.65 937
.34 063
.04 109
.38 172
28
33
.61 856
.95 885
.65 971
.34 029
.04115
.38 144
27
34
.61 883
.95 879
.66 004
.33 996
.04 121
.38 117
26
35
9.61 911
9.95 873
9.66 038
0.33 962
0.04 127
0.38 089
25
36
.61 939
.95 868
.66 071
.33 929
.04 132
.38 061
24
37
.61 966
.95 862
.66 104
.33 896
.04 138
.38 034
23
38
.61 994
.95 856
.66 138
.33 862
.04 144
.38 006
22
39
.62 021
.95850
.66 171
.33 829
.04 150
.37 979
21
40
9.62 049
9.95 844
9.66 204
0.33 796
0.04 156
0.37 951
20
41
.62 076
.95 839
.66 238
.33 762
.04 161
.37 924
19
42
.62 104
.95833
.66 271
.33 729
.04 167
.37 896
18
43
.62 131
.95 827
.66 304
.33 696
.04173
.37 869
17
44
.62 159
.95 821
.66 337
.33 663
.04 179
.37 841
16
45
9.62 186
9.95 815
9.66 371
0.33 629
0.04 185
0.37 814
15
46
.62 214
.95 810
.66 404
.33 596
.04 190
.37 786
14
47
.62 241
.95804
.66 437
.33 563
.04 196
.37 759
13
48
.62 268
.95 798
.66 470
.33 530
'.04 202
.37 732
12
49
.62 296
.95 792
.66 503
.33 497
.04 208
.37 704
11
50
9.62 323
9.95 786
9.66 537
0.33 463
0.04 214
0.37 677
10
51
.62 350
.95 780
.66 570
.33 430
.04220
.37 650
9
52
.62 377
.95 775
.66 603
.33 397
.04225
.37 623
8
53
.62 405
.95 769
.66 636
.33 364
.04231
.37 595
7
54
.62 432
.95 763
.66 669
.33 331
.04 237
.37 568
6
55
9.62 459
9.95 757
9.66 702
0.33 298
0.04 243
0.37 541
5
56
.62 486
.95 751
.66 735
.33 265
.04 249
.37 514
4
57
.62 513
.95 745
.66 768
.33 232
.04 255
.37 487
3
58
.62 541
.95 739
.66801
.33 199
.04261
.37 459
2
59
.62 568
.95 733
.66 834
.33 166
.04 267
.37 432
1
60
9.62 595
9.95 728
9.66 867
0.33 133
0.04 272
0.37 405
0
Cos
Sin
Cot
Tan
Csc
Sec
'
114° (294°)
(245°) 65°
Table 4. Trigonometric Logarithms
221
25° (205°)
(334°) 154°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.62 595
9.95 728
9.66 867
0.33 133
0.04 272
0.37 405
60
1
.62 622
.95 722
.66 900
.33 100
.04278
.37 378
59
2
.62649
.95 716
.66 933
.33 067
.04284
.37 351
58
3
.62 676
.95 710
.66 966
.33 034
.04290
.37 324
57
4
.62 703
.95 704
.66999
.33 001
.04296
.37 297
56
5
9.62 730
9.95 698
9.67 032
0.32 968
0.04 302
0.37 270
55
6
.62 757
.95 692
.67 065
.32 935
.04 308
.37 243
54
7
.62784
.95 686
.67 098
.32 902
.04 314
.37 216
53
8
.62811
.95 680
.67 131
.32 869
.04 320
.37 189
52
9
.62838
.95 674
.67 163
.32 837
.04 326
.37 162
51
10
9.62 865
9.95 668
9.67 196
0.32 804
0.04 332
0.37 135
50
11
.62 892
.95 663
.67 229
.32 771
.04 337
.37 108
49
12
.62 918
.95 657
.67 262
.32 738
.04343
.37 082
48
13
.62 945
.95 651
.67 295
.32 705
.04 349
.37 055
47
14
.62 972
.95 645
.67 327
.32 673
.04 355
.37 028
46
15
9.62 999
9.95 639
9.67 360
0.32 640
0.04 361
0.37 001
45
16
.63 026
.95 633
.67 393
.32 607
.04 367
.36 974
44
17
.63 052
.95 627
.67 426
.32 574
.04 373
.36 948
43
18
.63 079
.95 621
.67 458
.32 542
.04379
.36 921
42
19
.63 106
.95 615
.67 491
.32 509
.04 385
.36 894
41
20
9.63 133
9.95 609
9.67 524
0.32 476
0.04 391
0.36 867
40
21
.63 159
.95 603
.67 556
.32 444
.04 397
.36 841
39
22
.63 186
.95 597
.67 589
.32411
.04 403
.36 814
38
23
.63 213
.95 591
.67 622
.32 378
.04 409
.36 787
37
24
.63 239
.95 585
.67 654
.32 346
.04415
.36 761
36
25
9.63 266
9.95 579
9.67 687
0.32 313
0.04 421
0.36 734
35
26
.63 292
.95 573
.67 719
.32 281
.04 427
.36 708
34
27
.63 319
.95 567
.67 752
.32 248
.04 433
.36 681
33
28
.63 345
.95 561
.67 785
.32 215
.04 439
.36 655
32
29
.63 372
.95 555
.67 817
.32 183
.04445
.36 628
31
30
9.63 398
9.95 549
9.67 850
0.32 150
0.04 451
0.36 602
30
31
.63 425
.95 543
.67 882
.32 118
.04 457
.36 575
29
32
.63 451
.95 537
.67 915
.32 085
.04 463
.36 549
28
33
.63 478
.95 531
.67 947
.32 053
.04 469
.36 522
27
34
.63 504
.95 525
.67 980
.32 020
.04 475
.36 496
26
35
9.63 531
9.95 519
9.68 012
0.31 988
0.04 481
0.36 469
25
36
.63 557
.95 513
.68 044
.31 956
.04 487
.36 443
24
37
.63583
.95 507
.68 077
.31 923
.04 493
.36 417
23
38
.63 610
.95500
.68 109
.31 891
.04 500
.36 390
22
39
.63 636
.95 494
.68 142
.31 858
.04 506
.36 364
21
40
9.63 662
9.95 488
9.68 174
0.31 826
0.04 512
0.36 338
20
41
.63 689
.95 482
.68 206
.31 794
.04 518
. 36311
19
42
.63 715
.95 476
.68 239
.31 761
.04 524
.36 285
18
43
.63 741
.95 470
.68 271
.31 729
.04530
.36 259
17
44
.63 767
.95 464
.68 303
.31 697
.04536
.36 233
16
45
9.63 794
9.95 458
9.68 336
0.31 664
0.04 542
0.36 206
15
46
.63 820
.95 452
.68 368
.31 632
.04548
.36 180
14
47
.63846
.95 446
.68 400
.31 600
.04 554
.36 154
13
48
.63 872
.95 440
.68 432
.31 568
.04 560
.36 128
12
49
.63 898
.95 434
.68 465
.31 535
.04 566
.36 102
11
50
9.63 924
9.95 427
9.68 497
0.31 503
0.04 573
0.36 076
10
51
.63 950
.95 421
.68 529
.31 471
.04 579
.36 050
9
52
.63 976
.95 415
.68 561
.31 439
.04585
.36 024
8
53
.64002
.95 409
.68 593
.31 407
.04591
.35 998
7
54
.64028
.95 403
.68 626
.31 374
.04597
.35 972
6
55
9.64 054
9.95 397
9.68 658
0.31 342
0.04 603
0.35 946
5
56
.64080
.95 391
.68 690
.31 310
.04609
.35 920
4
57
.64 106
.95 384
.68 722
.31 278
.04 616
.35 894
3
58
.64 132
.95 378
' .68754
.31 246
.04 622
.35 868
2
59
.64 158
.95 372
.68 786
.31 214
.04 628
.35842
1.
60
9.64 184
9.95 366
9.68 818
0.31 182
0.04 634
0.35 816
0
Cos
Sin
Cot
Tan
Csc
Sec
'
115° (295°)
(244°) 64°
222
Table 4. Trigonometric Logarithms
26° (206°)
(333°) 153°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.64 184
9.95 366
9.68 818
0.31 182
0.04 634
0.35 816
60
1
.64 210
.95 360
.68 850
.31 150
.04 640
.35 790
59
2
.64 236
.95 354
.68 882
.31 118
.04 646
.35 764
58
3
.64262
.95 348
.68 914
.31 086
.04 652
.35 738
57
4
.64288
.95 341
.68 946
.31054
.04659
.35 712
56
5
9.64 313
9.95 335
9.68 978
0.31 022
0.04 665
0.35 687
55
6
.64 339
.95 329
.69 010
.30 990
.04 671
.35 661
54
7
.64365
.95 323
.69 042
.30 958
.04677
.35 635
53
8
.64391
.95 317
.69 074
.30 926
.04 683
.35 609
52
9
.64417
.95 310
.69 106
.30 894
.04690
.35 583
51
10
9.64 442
9.95 304
9.69 138
0.30 862
0.04 696
0.35 558
50
11
.64 468
.95 298
.69 170
.30 830
.04702
.35 532
49
12
.64 494
.95 292
.69 202
.30 798
.04 708
.35 506
48
13
.64 519
.95 286
.69 234
.30 766
.04 714
.35 481
47
14
.64 545
.95 279
.69 266
.30 734
.04 721
.35 455
46
15
9.64 571
9.95 273
9.69 298
0.30 702
0.04 727
0.35 429
45
16
.64 596
.95 267
.69 329
.30 671
.04 733
.35 404
44
17
.64622
.95 261
.69 361
.30 639
.04739
.35 378
43
18
.64647
.95 254
.69 393
.30 607
.04746
.35 353
42
19
.64673
.95 248
.69 425
.30 575
.04 752
.35 327
41
20
9.64 698
9.95 242
9.69 457
0.30 543
0.04 758
0.35 302
40
21
.64724
.95 236
.69 488
.30 512
.04 764
.35 276
39
22
.64 749
.95 229
.69 520
.30 480
.04 771
.35 251
38
23
.64 775
.95 223
.69 552
.30 448
.04 777
.35 225
37
24
.64800
.95 217
.69584
.30 416
.04 783
.35 200
36
25
9.64 826
9.95211
9.69 615
0.30 385
0.04 789
0.35 174
35
26
.64851
.95 204
.69 647
.30 353
.04 796
.35 149
34
27
.64877
.95 198
.69 679
.30 321
.04802
.35 123
33
28
.64 902
.95 192
.69 710
.30 290
.04808
.35 098
32
29
.64927
.95 185
.69 742
.30 258
.04815
.35 073
31
30
9.64 953
9.95 179
9.69 774
0.30 226
0.04 821
0.35 047
30
31
.64978
.95 173
.69 805
.30 195
.04 827
.35 022
29
32
.65 003
.95 167
.69 837
.30 163
.04 833
.34 997
.28
33
.65 029
.95 160
.69 868
.30 132
.04840
.34 971
27
34
.65 054
.95 154
.69 900
.30 100
.04846
.34 946
26
35
9.65 079
9.95 148
9.69 932
0.30 068
0.04 852
0.34 921
25
36
.65 104
.95 141
.69 963
.30 037
.04 859
.34 896
24
37
.65 130
.95 135
.69 995
.30 005
.04 865
.34 870
23
38
.65 155
.95 129
.70 026
.29 974
.04 871
.34845
22
39
.65 180
.95 122
.70 058
.29 942
.04 878
.34 820
21
40
9.65 205
9.95 116
9.70 089
0.29911
0.04884
0.34 795
20
41
.65 230
.95 110
.70 121
.29 879
.04 890
.34 770
19
42
.65 255
.95 103
.70 152
.29848
.04 897
.34 745
18
43
.65 281
.95 097
.70 184
.29 816
.04 903
.34 719
17
44
.65 306
.95 090
.70 215
.29 785
.04 910
.34 694
16
45
9.65 331
9.95 084
9.70 247
0.29 753
0.04 916
0.34 669
15
46
.65 356
.95 078
.70 278
.29 722
.04 922
.34 644
14
47
.65 381
.95 071
.70 309
.29 691
.04 929
..°4 619
13
48
.65 406
.95 065
.70 341
.29 659
.04 935
.34 594
12
49
.65 431
.95 059
.70 372
.29 628
.04 941
.34 569
11
50
9.65 456
9.95 052
9.70 404
0.29 596
0.04 948
0.34 544
10
51
.65 481
.95 046
.70 435
.29 565
.04 954
.34 519
9
52
.65 506
.95 039
.70 466
.29 534
.04 961
.34 494
8
53
.65 531
.95 033
.70 498
.29 502
.04 967
.34 469
7
54
.65 556
.95 027
.70 529
.29 471
.04 973
.34 444
6
55
9.65 580
9.95 020
9.70 560
0.29 440
0.04 980
0.34 420
5
56
.65 605
.95 014
.70 592
.29 408
.04 986
.34 395
4
57
.65 630
.95 007
.70 623
.29 377
.04 993
.34 370
3
58
.65 655
.95 001
.70 654
.29 346
.04 999
.34 345
2
59
.65 680
.94 995
.70 685
.29 315
.05 005
.34 320
1
60
9.65 705
9.94 988
9.70 717
0.29 283
0.05 012
0.34 295
0
Cos
Sin
Cot
Tan
Csc
Sec
'
116° (296°)
(243°) 63°
Table 4. Trigonometric Logarithms
223
27° (207°)
(332°) 152°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.65 705
9.94 988
9.70717
0.29 283
0.05 012
0.34 295
60
1
.65 729
.94 982
.70 748
.29 252
.05 018
.34 271
59
2
.65 754
.94 975
.70 779
.29 221
.05 025
.34 246
58
3
.65 779
.94 969
.70 810
.29 190
.05 031
.34 221
57
4 .
.65804
.94 962
.70841
.29 159
.05 038
.34 196
56
5
9.65 828
9.94 956
9.70 873
0.29 127
0.05044
0.34 172
55
6
.65853
.94 949
.70904
.29 096
.05 051
.34 147
54
7
.65 878
.94943
.70 935
.29 065
.05 057
.34 122
53
8
.65 902
.94 936
.70 966
.29 034
.05064
.34 098
52
9
.65 927
.94 930
.70 997
.29 003
.05 070
.34 073
51
10
9.65 952
9.94 923
9.71 028
0.28 972
0.05 077
0.34 048
50
11
.65 976
.94917
.71 059
.28 941
.05 083
.34 024
49
12
.66001
.94911
.71 090
.28 910
.05 089
.33 999
48
13
.66 025
.94904
.71 121
.28 879
.05 096
.33 975
47
14
.66 050
.94 898
.71 153
.28847
.05 102
.33 950
46
15
9.66 075
9.94 891
9.71 184
0.28 816
0.05 109
0.33 925
45
16
.66 099
.94 885
.71 215
.28 785
.05 115
.33 901
44
17
.66 124
.94 878
.71 246
.28754
.05 122
.33 876
43
18
.66 148
.94 871
.71 277
.28 723
.05 129
.33 852
42
19
.66 173
.94 865
.71 308
.28 692
.05 135
.33 827
41
20
9.66 197
9.94 858
9.71 339
0.28 661
0.05 142
0.33 803
40
21
.66 221
.94852
.71 370
.28 630
.05 148
.33 779
39
22
.66 246
.94845
.71 401
.28 599
.05 155
.33 754
38
23
.66270
.94839
.71 431
.28 569
.05 161
.33 730
37
24
.66 295
.94832
.71 462
.28 538
.05 168
.33 705
36
25
9.66 319
9.94 826
9.71 493
0.28 507
0.05 174
0.33 681
35
26
.66 343
.94 819
.71 524
.28 476
.05 181
.33 657
34
27
.66 368
.94 813
.71 555
.28445
.05 187
.33 632
33
28
.66 392
.94806
.71 586
.28.414
.05 194
.33 608
32
29
.66416
.94 799
.71 617
.28383
.05 201
.33584
31
30
9.66 441
9.94 793
9.71 648
0.28 352
0.05 207
0.33 559
30
31
.66 465
.94 786
.71 679
.28 321
.05 214
.33 535
29
32
.66 489
.94 780
.71 709
.28 291
.05 220
.33511
28
33
.66 513
.94 773
.71 740
.28 260
.05 227
.33 487
27
34
.66 537
.94 767
.71 771
.28 229
.05 233
.33 463
26
35
9.66 562
9.94 760
9.71 802
0.28 198
0.05 240
0.33 438
25
36
.66 586
.94 753
.71833
.28 167
.05 247
.33 414
24
37
.66 610
.94 747
.71 863
.28 137
.05 253
.33 390
23
38
.66 634
.94 740
.71 894
.28 106
.05 260
.33 366
22
39
.66658
.94 734
.71 925
.28 075
.05 266
.33 342
21
40
9.66 682
9.94 727
9.71 955
0.28 045
0.05 273
0.33 318
20
41
.66 706
.94 720
.71 986
.28 014
.05 280
.33 294
19
42
.66 731
.94 714
.72 017
.27 983
.05 286
.33 269
18
43
.66 755
.94 707
.72048
.27 952
.05 293
.33 245
17
44
.66 779
.94700
.72 078
.27 922
.05 300
.33 221
16
45
9.66 803
9.94 694
9.72 109
0.27 891
0.05 306
0.33 197
15
46
.66 827
.94 687
.72 140
.27860
.05 313
.33 173
14
47
.66851
.94 680
.72 170
.27830
.05 320
.33 149
13
48
.66875
.94674
.72 201
.27 799
.05 326
.33 125
12
49
.66 899
.94 667
.72 231
.27 769
.05 333
.33 101
11
50
9.66 922
9.94 660
9.72 262
0.27 738
0.05 340
0.33 078
10
51
.66 946
.94654
.72 293
.27 707
.05 346
.33 054
9
52
.66970
.94647
.72 323
.27 677
.05 353
.33 030
8
53
.66994
.94640
.72 354
.27646
.05 360
.33006
7
54
.67 018
.94 634
.72384
.27 616
.05366
.32 982
6
55
9.67 042
9.94 627
9.72 415
0.27 585
0.05 373
0.32 958
5
56
.67 066
.94 620
.72445
.27 555
.05380
.32 934
4
57
.67090
.94 614
.72 476
.27 524
.05386
.32 910
3
58
.67 113
.94607
.72 506
.27 494
.05 393
.32887
2
59
.67 137
.94600
.72 537
.27 463
.05 400
.32 863
1
60
9.67 161
9.94 593
9.72 567
0.27 433
0.05 407
0.32 839
0
Cos
Sin
Cot
T;in
Csc
Sec
'
117° (297°)
(242°) 62°
224
Table 4. Trigonometric Logarithms
28° (208°)
(331°) 151°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.67 161
9.94 593
9.72 567
0.27 433
0.05 407
0.32 839
60
1
.67 185
.94 587
.72 598
.27 402
.05413
.32 815
59
2
.67 208
.94 580
.72 628
.27 372
.05 420
.32 792
58
3
.67 232
.94 573
.72 659
.27 341
.05 427
.32 768
57
4
.67 256
.94 567
.72 689
.27311
.05 433
.32 744
56
5
9.67 280
9.94 560
9.72 720
0.27 280
0.05 440
0.32 720
55
6
.67 303
.94 553
.72 750
.27 250
.05 447
.32 697
54
7
.67 327
.94 546
.72 780
.27 220
.05 454
.32 673
53
8
.67 350
.94 540
.72 811
.27 189
.05 460
.32 650
52
9
.67 374
.94 533
.72841
.27 159
.05 467
.32 626
51
10
9.67 398
9.94 526
9.72 872
0.27 128
0.05 474
0.32 602
50
11
.67 421
.94 519
.72 902
.27 098
.05 481
.32 579
49
12
.67 445
.94 513
.72 932
.27 068
.05 487
.32 555
48
13
.67 468
.94 506
.72 963
.27 037
.05 494
.32 532
47
14
.67 492
.94 499
.72 993
.27 007
.05 501
.32 508
46
15
9.67 515
9.94 492
9.73 023
0.26 977
0.05 508
0.32 485
45
16
.67 539
.94 485
.73 054
.26 946
.05 515
.32 461
44
17
.67 562
.94 479
.73084
.26 916
.05 521
.32 438
43
18
.67 586
.94 472
.73 114
.26 886
.05 528
.32 414
42
19
.67 609
.94 465
.73 144
.26 856
.05 535
.32 391
41
20
9.67 633
9.94 458
9.73 175
0.26 825
0.05 542
0.32 367
40
21
.67 656
.94 451
.73 205
.26 795
.05 549
.32 344
39
22
.67 680
.94 445
.73 235
.26 765
.05 555
.32 320
38
23
.67 703
.94 438
.73 265
.26 735
.05 562
.32 297
37
24
.67 726
.94 431
.73 295
.26 705
.05 569
.32 274
36
25
9.67 750
9.94 424
9.73 326
0.26 674
0.05 576
0.32 250
35
26
.67 773
.94417
.73 356
.26 644
.05 583
.32 227
34
27
.67 796
.94 410
.73 386
.26 614
.05 590
.32 204
33
28
.67 820
.94 404
.73 416
.26 584
.05 596
.32 180
32
29
.67843
.94 397
.73 446
.26 554
.05 603
.32 157
31
30
9.67 866
9.94 390
9.73 476
0.26 524
0.05 610
0.32 134
30
31
.67 890
.94 383
.73 507
.26 493
.05 617
.32 110
29
32
.67 913
.94 376
.73 537
.26 463
.05 624
.32 087
28
33
.67 936
.94 369
.73 567
.26 433
.05 631
.32 064
27
34
.67 959
.94 362
.73 597
.26 403
.05 638
.32 041
26
35
9.67 982
9.94 355
9.73 627
0.26 373
0.05 645
0.32 018
25
36
.68 006
.94 349
.73 657
.26 343
.05 651
.31 994
24
37
.68 029
.94 342
.73 687
.26 313
.05 658
.31 971
23
38
.68 052
.94 335
.73 717
.26 283
.05 665
.31 948
22
39
.68 075
.94 328
.73 747
.26 253
.05 672
.31 925
21
40
9.68 098
9.94 321
9.73 777
0.26 223
0.05 679
0.31 902
20
41
.68 121
.94 314
.73 807
.26 193
.05 686
.31 879
19
42
.68 144
.94 307
.73 837
.26 163
.05 693
.31 856
18
43
.68 167
.94 300
.73 867
.26 133
.05 700
.31 833
17
44
.68 190
.94 293
.73 897
.26 103
.05 707
.31 810
16
45
9.68 213
9.94 286
9.73 927
0.26 073
0.05 714
0.31 787
15
46
.68 237
.94 279
.73 957
.26 043
.05 721
.31 763
14
47
.68 260
.94 273
.73 987
.26 013
.05 727
.?! 740
13
48
.68 283
.94 266
.74 017
.25 983
.05 734
.31 717
12
49
.68 305
.94 259
.74 047
.25 953
.05 741
.31 695
11
50
9.68 328
9.94 252
9.74 077
0.25 923
0.05 748
0.31 672
10
51
.68 351
.94 245
.74 107
.25 893
.05 755
.31 649
9
52
.68 374
.94 238
.74 137
.25 863
.05 762
.31 626
8
53
.68 397
.94 231
.74 166
.25 834
.05 769
.31 603
7
54
.68 420
.94 224
.74 196
.25 804
.05 776
.31 580
6
55
9.68 443
9.94 217
9.74 226
0.25 774
0.05 783
0.31 557
5
56
.68 466
.94 210
.74 256
.25 744
.05 790
.31 534
4
57
.68 489
.94 203
.74 286
.25 714
.05 797
.31 511
3
58
.68512
.94 196
.74 316
.25 684
.05804
.31 488
2
59
.68 534
.94 189
.74 345
.25 655
.05811
.31 466
1
60
9.68 557
9.94 182
9.74 375
0.25 625
0.05 818
0.31 443
0
Cos
Sin
Cot
Tan
Csc
Sec
'
118° (298°)
(241°) 61C
Table 4. Trigonometric Logarithms
225
29° (209°)
(330°) 150°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.68 557
9.94 182
9.74 375
0.25 625
0.05 818
0.31 443
60
1
.68 580
.94 175
.74 405
.25 595
.05 825
.31 420
59
2
.68 603
.94 168
.74 435
.25 565
.05832
.31 397
58
3
.68 625
.94 161
.74 465
.25 535
.05839
.31 375
57
4
.68648
.94154
.74 494
.25506
.05846
.31 352
56
5
9.68 671
9.94 147
9.74 524
0.25 476
0.05 853
0.31 329
55
6
.68 694
.94 140
.74 554
.25446
.05 860
.31 306
54
7
.68 716
.94 133
.74 583
.25 417
.05 867
.31 284
53
8
.68 739
.94 126
.74 613
.25 387
.05 874
.31 261
52
9
.68 762
.94 119
.74643
.25 357
.05 881
.31 238
51
10
9.68 784
9.94 112
9.74 673
0.25 327
0.05 888
0.31 216
50
11
.68807
.94 105
.74 702
.25 298
.05 895
.31 193
49
12
.68 829
.94 098
.74 732
.25 268
.05 902
.31 171
48
13
.68852
.94 090
.74 762
.25 238
.05 910
.31 148
47
14
.68 875
.94083
.74 791
.25 209
.05 917
.31 125
46
15
9.68 897
9.94 076
9.74 821
0.25 179
0.05 924
0.31 103
45
16
.68 920
.94 069
.74 851
.25 149
.05 931
.31 080
44
17
.68 942
.94 062
.74 880
.25 120
.05 938
.31 058
43
18
.68 965
.94 055
.74 910
.25 090
.05 945
.31 035
42
19
.68 987
.94048
.74 939
.25 061
.05 952
.31 013
41
20
9.69 010
9.94 041
9.74 969
0.25 031
0.05 959
0.30 990
40
21
.69 032
.94 034
.74 998
.25 002
.05 966
.30 968
39
22
.69 055
.94 027
.75 028
.24 972
.05 973
.30 945
38
23
.69 077
.94 020
.75 058
.24 942
.05 980
.30 923
37
24
.69 100
.94 012
.75 087
.24 913
.05 988
.30 900
36
25
9.69 122
9.94 005
9.75 117
0.24 883
0.05 995
0.30 878
35
26
.69 144
.93 998
.75 146
.24 854
.06002
.30 856
34
27
.69 167
.93 991
.75 176
.24 824
.06 009
.30 833
33
28
i69 189
.93984
.75 205
.24 795
.06 016
.30811
32
29
.69 212
.93 977
.75 235
.24 765
.06023
.30 788
31
30
9.69 234
9.93 970
9.75 264
0.24 736
0.06 030
0.30 766
30
31
.69 256
.93 963
.75 294
.24 706
.06 037
.30744
29
32
.69 279
.93 955
.75 323
.24 677
.06045
.30 721
28
33
.69 301
• .93948
.75 353
.24647
.06 052
.30 699
27
34
.69 323
.93 941
.75 382
.24618
.06 059
.30 677
26
35
9.69 345
9.93 934
9.75411
0.24 589
0.06 066
0.30 655
25
36
.69 368
.93 927
.75 441
.24 559
.06073
.30 632
24
37
.69 390
.93 920
.75 470
.24530
.06 080
.30 610
23
38
.69 412
.93 912
.75500
.24 500
.06 088
.30588
22
39
.69 434
.93 905
.75 529
.24 471
.06 095
.30 566
21
40
9.69 456
9.93 898
9.75 558*
0.24 442
0.06 102
0.30 544
20
41
.69 479
.93 891
.75 588
.24 412
.06 109
.30 521
19
42
.69 501
.93884
.75 617
.24 383
.06 116
.30 499
18
43
.69 523
.93 876
.75647
.24 353
.06 124
.30 477
17
44
.69545
.93 869
.75 676
.24 324
.06 131
.30 455
16
45
9.69 567
9.93 862
9.75 705
0.24 295
0.06 138
0.30 433
15
46
.69 589
.93855
.75 735
.24 265
.06 145
.30411
14
47
.69611
.93847
.75764
.24 236
.06 153
.30 389
13
48
.69 633
.93840
.75 793
.24 207
.06 160
.30 367
12
49
.69 655
..93 833
.75 822
.24 178
.06167
.30 345
11
50
9.69 677
9.93 826
9.75 852
0.24 148
0.06 174
0.30 323
10
51
.69 699
.93 819
.75 881
.24 119
.06 181
.30 301
9
52
.69 721
.93811
.75 910
.24 090
.06 189
.30 279
8
53
.69 743
.93804
.75 939
.24061
.06 196
.30 257
7
54
.69 765
.93 797
.75 969
.24 031
.06 203
.30 235
6
55
9.69 787
9.93 789
9.75 998
0.24 002
0.06211
0.30 213
5
56
.69809
.93 782
.76 027
.23 973
.06 218
.30 191
4
57
.69831
.93 775
.76 056
.23944
.06 225
.30 169
3
58
.69853
.93 768
.76 086
.23 914
.06 232
.30 147
2
59
.69 875
.93 760
.76 115
.23 885
.06 240
.30 125
1
60
9.69 897
9.93 753 9.76 144
0.23 856
0.06 247
0.30 103
0
Cos
Sin Cot
T;in
Csc
Sec
'
119° (299°)
(240°) 60°
226
Table 4. Trigonometric Logarithms
30° (210°)
(329°) 149°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.69 897
9.93 753
9.76 144
0.23 856
0.06 247
0.30 103
60
1
.69 919
.93 746
.76 173
.23 827
.06 254
.30 081
59
2
.69 941
.93 738
.76 202
.23 798
.06 262
.30 059
58
3
.69 963
.93 731
.76 231
.23 769
.06 269
.30 037
57
4
.69984
.93 724
.76 261
.23 739
.06 276
.30 016
56
5
9.70 006
9.93 717
9.76 290
0.23 710
0.06 283
0.29 994
55
6
.70 028
.93 709
.76 319
.23 681
.06 291
.29 972
54
7
.70 050
.93 702
.76 348
.23 652
.06 298
.29 950
53
8
.70 072
.93 695
.76 377
.23 623
.06 305
.29 928
52
9
.70 093
.93 687
.76 406
.23 594
.06 313
.29 907
51
10
9.70 115
9.93 680
9.76 435
0.23 565
0.06 320
0.29 885
50
11
.70 137
.93 673
.76464
.23 536
.06 327
.29 863
49
12
.70 159
.93 665
.76 493
.23 507
.06 335
.29841
48
13
.70 180
.93 658
.76 522
.23 478
.06 342
.29 820
45
14
.70 202
.93 650
.76 551
.23 449
.06 350
.29 798
46
15
9.70 224
9.93 643
9.76 580
0.23 420
0.06 357
0.29 776
45
16
.70 245
.93 636
.76 609
.23 391
.06 364
.29 755
44
17
.70 267
.93 628
.76 639
.23 361
.06 372
.29 733
43
18
.70 288
.93 621
.76 668
.23 332
.06 379
.29 712
42
19
.70 310
.93 614
.76 697
.23 303
.06 386
.29 690
41
20
9.70 332
9.93 606
9.76 725
0.23 275
0.06 394
0.29 668
40
21
.70 353
.93 599
.76 754
.23 246
.06 401
.29 647
39
22
.70 375
.93 591
.76 783
.23 217
.06 409
.29 625
38
23
.70 396
.93584
.76 812
.23 188
.06 416
.29 604
37
24
.70 418
.93 577
.76841
.23 159
.06 423
.29 582
36
25
9.70 439
9.93 569
9.76 870
0.23 130
0.06 431
0.29 561
35
26
.70 461
.93 562
.76 899
.23 101
.06 438
.29 539
34
27
.70 482
.93 554
.76 928
.23 072
.06 446
.29 518
33
28
.70 504
.93 547
.76 957
.23 043
.06 453
.29 496
32
29
.70 525
.93 539
.76 986
.23 014
.06 461
.29 475
31
30
9.70 547
9.93 532
9.77 015
0.22 985
0.06 468
0.29 453
30
31
.70 568
.93 525
.77 044
.22 956
.06 475
.29 432
29
32
.70 590
.93 517
.77 073
.22 927
.06 483
.29 410
28
33
.70611
.93 510
.77 101
.22 899
.06 490
.29 389
27
34
.70 633
.93 502
.77 130
.22 870
.06 498
.29 367
26
35
9.70 654
9.93 495
9.77 159
0.22 841
0.06 505
0.29 346
25
36
.70 675
.93 487
.77 188
.22 812
.06 513
.29 325
24
37
.70 697
.93 480
.77 217
.22 783
.06 520
.29 303
23
38
.70 718
.93 472
.77 246
.22 754
.06 528
.29 282
22
39
.70 739
.93 465
.77 274
.22 726
.06 535
.29 261
21
40
9.70 761
9.93 457
9.77 303 "
0.22 697
0.06 543
0.29 239
20
41
.70 782
.93 450
.77 332
.22 668
.06 550
.29218
19
42
.70 803
.93 442
.77 361
.22 639
.06 558
.29 197
18
43
.70 824
.93 435
.77 390
.22 610
.06 505
.29 176
17
44
.70 846
.93 427
.77 418
.22 582
.06 573
.29 154
16
45
9.70 867
9.93 420
9.77 447
0.22 553
0.06 580
0.29 133
15
46
.70 888
.93 412
.77 476
.22 524
.06 588
.29 112
14
47
.70 909
.93 405
.77 505
.22 495
.06 595
29091
13
48
.70 931
.93 397
.77 533
.22 467
.06 603
.29 069
12
49
.70 952
.93 390
.77 562
.22 438
.06 610
.29 048
11
50
9.70 973
9.93 382
9.77 591
0.22 409
0.06 618
0.29 027
10
51
.70 994
.93 375
.77 619
.22 381
.06 625
.29 006
9
52
.71 015
.93 367
.77 648
.22 352
.06 633
.28 985
8
53
.71 036
.93 360
.77 677
.22 323
.06 640
.28964
7
54
.71 058
.93 352
.77 706
.22 294
.06 648
.28 942
6
55
9.71 079
9.93 344
9.77 734
0.22 266
0.06 656
0.28 921
5
56
.71 100
.93 337
.77 763
.22 237
.06 663
.28 900
4
57
.71 121
.93 329
.77 791
.22 209
.06 671
.28 879
3
58
.71 142
.93 322
.77 820
.22 180
.06 678
.28 858
2
59
.71 163
.93 314
.77 849
.22 151
.06 686
.28837
1
60
9.71 184
9.93 307
9.77 877
0.22 123
0.06 693
0.28816
0
Cos
Sin
Cot
Tan
Csc
Sec
'
120° (300°)
(239°) 59°
Table 4. Trigonometric Logarithms
227
31° (211°)
(328°) 148°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.71 184
9.93 307
9.77 877
0.22 liJ3
0.06 693
0.28 816
60
1
.71 205
.93 299
.77 906
.22 094
.06 701
.28 795
59
2
.71 226
.93 291
.77 935
.22065
.06 709
.28 774
58
3
.71 247
.93284
.77 963
.22 037
.06 716
.28 753
57
4
.71 268
.93 276
.77 992
.22008
.06724
.28 732
56
5
9.71 289
9.93 269
9.78 020
0.21 980
0.06 731
0.28 711
55
6
.71 310
.93 261
.78 049
.21 951
.06 739
.28 690
54
7
.71 331
.93 253
.78 077
.21 923
.06 747
.28 669
53
8
.71 352
.93 246
.78 106
'.21 894
.06754
.28 648
52
9
.71 373
.93 238
.78 135
.21 865
.06762
.28 627
51
10
9.71 393
9.93 230
9.78 163
0.21 837
0.06 770
0.28 607
50
11
.71 414
.93 223
.78 192
.21 808
.06 777
.28 586
49
12
.71 435
.93 215
.78 220
.21 780
.06785
.28 565
48
13
.71 456
.93 207
.78 249
.21 751
.06 793
.28 544
47
14
.71 477
.93 200
.78 277
.21 723
.06 800
.28 523
46
15
9.71 498
9.93 192
9.78 306
0.21 694
0.06 808
0.28 502
45
16
.71 519
.93 184
.78 334
.21 666
.06 816
.28 481
44
17
.71 539
.93 177
.78 363
.21 637
.06823
.28 461
43
18
.71 560
.93 169
.78 391
.21 609
.06 831
.28440
42
19
.71 581
.93 161
.78 419
.21 581
.06839
.28 419
41
20
9.71 602
9.93 154
.78 448
0.21 552
0.06 846
0.28 398
40
21
.71 622
.93 146
.78 476
.21 524
.06 854
.28 378
39
22
.71 643
.93 138
.78 505
.21 495
.06 862
.28 357
38
23
.71664
.93 131
.78 533
.21 467
.06 869
.28 336
37
24
.71 685
.93 123
.78 562
.21 438
.06877
.28 315
36
25
9.71 705
9.93 115
9.78 590
0.21 410
0.06 885
0.28 295
35
26
.71 726
.93 108
.78 618
.21 382
.06 892
.28 274
34
27
.71 747
.93 100
.78647
.21 353
.06 900
.28 253
33
28
.71 767
.93 092
.78 675
.21 325
.06 908
.28 233
32
29
.71 788
.93084
.78 704
.21 296
.06 916
.28 212
31
30
9.71 809
9.93 077
9.78 732
0.21 268
0.06 923
0.28 191
30
31
.71 829
.93 069
.78 760
.21 240
.06931
.28 171
29
32
.71850
.93 061
.78 789
.21 211
.06 939
.28 150
28
33
.71 870
.93 053
.78 817
.21 183
.06 947
.28 130
27
34
.71 891
.93046
.78845
.21 155
.06 954
.28 109
26
35
9.71911
9.93 038
9.78 874
0.21 126
0.06 962
0.28 089
25
36
.71 932
.93 030
.78 902
.21 098
.06 970
.28 068
24
37
.71 952
.93 022
.78 930
.21 070
.06 978
.28048
23
38
.71 973
.93 014
.78 959
.21 041
.06 986
.28 027
22
39
.71 994
.93 007
.78 987
.21 013
.06 993
.28 006
21
40
9.72 014
9.92 999
9.79 015
0.20 985
0.07 001
0.27 986
20
41
.72 034
.92 991
.79043
.20 957
.07009
.27 966
19
42
.72 055
.92 983
.79 072
.20 928
.07 017
.27 945
18
43
.72 075
.92 976
.79 100
.20900
.07 024
.27 925
17
44
.72 096
.92 968
.79 128
.20 872
.07 032
.27904
16
45
9.72 116
9.92 960
9.79 156
0.20844
0.07 040
0.27 884
15
46
.72 137
.92 952
.79 185
.20 815
.07 048
.27 863
14
47
.72 157
.92944
.79 213
.20 787
.07 056
.27843
13
48
.72 177
.92 936
.79 241
.20 759
.07 064
.27 823
12
49
.72 198
.92 929
.79 269
.20 731
.07 071
.27 802
11
50
9.72 218
9.92 921
9.79 297
0.20 703
0.07 079
0.27 782
10
51
.72 238
.92 913
.79 326
.20 674
.07 087
.27 762
9
52
.72 259
.92 905
.79 354
.20646
.07 095
.27 741
8
53
.72 279
.92 897
.79 382
.20 618
.07 103
.27 721
7
54
.72 299
.92889
.79 410
.20 590
.07 111
.27 701
6
55
9.72 320
9.92 881
9.79 438
0.20 562
0.07 119
0.27 680
5
56
.72 340
.92 874
.79 466
.20 534
.07 126
.27 660
4
57
.72 360
.92 866
.79 495
.20 505
.07134
.27640
3
58
.72 381
.92 858
.79 523
.20 477
.07 142
.27 619
2
59
.72 401
.92850
.79 551
.20 449
.07 150
.27 599
1
60
9.72 421
9.92 842
9.79 579
0.20 421
0.07 158
0.27 579
0
Cos
Sin
Cot
Tan
Csc
Sec
'
121° (301°)
(238°) 58°
228
Table 4. Trigonometric Logarithms
32° (212°)
(327°) 147°
/
Sin
Cos
Tan
Cot
Sec
Csc
0
9.72 421
9.92 842
9.79 579
0.20 421
0.07 158
0.27 579
60
1
.72 441
.92 834
.79 607
.20 393
.07 166
.27 559
59
2
.72 461
.92 826
.79 635
.20 365
.07 174
.27 539
58
3
.72 482
.92 818
.79 663
.20 337
.07 182
.27 518
57
4
.72 502
.92 810
.79 691
.20 309
.07 190
.27 498
56
5
9.72 522
9.92 803
9.79 719
0.20 281
0.07 197
0.27 478
55
6
.72 542
.92 795
.79 747
.20 253
.07 205
.27 458
54
7
.72 562
.92 787
.79 776
.20 224
.07 213
.27 438
53
8
.72 582
.92 779
.79 804
.20 196
.07 221
.27 418
52
9
.72 602
.92 771
.79 832
.20 168
.07 229
.27 398
51
10
9.72 622
9.92 763
9.79 860
0.20 140
0.07 237
0.27 378
50
11
.72643
.92 755
.79 888
.20 112
.07 245
.27 357
49
12
.72 663
.92 747
.79 916
.20084
.07 253
.27 337
48
13
.72 683
.92 739
.79 944
.20 056
.07 261
.27 317
47
14
.72 703
.92 731
.79 972
.20 028
.07 269
.27 297
46
15
9.72 723
9.92 723
9.80 000
0.20 000
0.07 277
0.27 277
45
16
.72 743
.92 715
.80 028
.19 972
.07 285
.27 257
44
17
.72 763
.92 707
.80 056
.19 944
.07 293
.27 237
43
18
•72 783
.92 699
.80084
.19916
.07 301
.27 217
42
19
.72 803
.92 691
.80112
.19 888
.07 309
.27 197
41
20
9.72 823
9.92 683
9.80 140
0.19 860
0.07 317
0.27 177
40
21
.72843
.92 675
.80 168
.19 832
.07 325
.27 157
39
22
.72 863
.92 667
.80 195
.19 805
.07 333
.27 137
38
23
.72 883
.92 659
.80 223
.19 777
.07 341
.27 117
37
24
.72 902
.92 651
.80 251
.19 749
.07 349
.27 098
36
25
9.72 922
9.92 643
9.80 279
0.19 721
0.07 357
0.27 078
35
26
.72 942
.92 635
.80 307
.19 693
.07 365
.27 058
34
27
.72 962
.92 627
.80 335
.19 665
.07 373
.27 038
33
28
.72 982
.92 619
.80 363
.19 637
.07 381
.27 018
32
29
.73 002
.92611
.80391
.19 609
.07389
.26 998
31
30
9.73 022
0.92 603
9.80 419
0.19 581
0.07 397
0.26 978
30
31
.73 041
.92 595
.80 447
.19 553
.07 405
.26 959
29
32
•73 061
.92 587
.80 474
.19 526
.07 413
.26 939
28
33
.73 081
.92 579
.80 502
.19 498
.07 421
.26 919
27
34
.73 101
.92 571
.80 530
.19 470
.07 429
.26 899
26
35
9.73 121
9.92 563
9.80 558
0.19 442
0.07 437
0.26 879
25
36
.73 140
.92 555
.80 586
.19414
.07 445
.26 860
24
37
.73 160
.92 546
.80 614
.19 386
.07 454
.26840
23
38
.73 180
.92 538
.80 642
.19 358
.07 462
.26 820
22
39
.73 200
.92 530
.80 669
.19331
.07 470
.26 800
21
40
9.73 219
9.92 522
9.80 697
0.19 303
0.07 478
0.26 781
20
41
.73 239
.92 514
.80 725
.19 275
.07 486
.26 761
19
42
.73 259
.92 506
.80 753
.19 247
.07 494
.26 741
18
43
.73 278
.92 498
.80781
.19219
.07 502
.26 722
17
44
.73 298
.92 490
.80 808
.19 192
.07 510
.26 702
16
45
9.73 318
9.92 482
9.80 836
0.19 164
0.07 518
0.26 682
15
46
.73 337
.92 473
.80 864
.19 136
.07 527
.26 663
14
47
.73 357
.92 465
.80 892
.19 108
.07 535
.26 643
13
48
.73 377
.92 457
.80919
.19 081
.07 543
.26 623
12
49
.73 396
.92 449
.80 947
.19 053
.07 551
.26 604
11
50
9.73416
9.92 441
9.80 975
0.19 025
0.07 559
0.26 584
10
51
.73 435
.92 433
.81 003
.18 997
.07 567
.26 565
9
52
.73 455
.92 425
.81 030
.18970
.07 575
.26 545
8
53
.73 474
.92 416
.81 058
.18 942
.07 584
.26 526
7
54
.73 494
.92 408
.81 086
.18914
.07 592
.26 506
6
55
9.73 513
9.92 400
9.81 113
0.18887
0.07 600
0.26 487
5
56
.73 533
.92 392
.81 141
.18 859
.07 608
.26 467
4
57
.73 552
.92 384
.81 169
.18831
.07 616
.26 448
3
58
.73 572
.92 376
.81 196
.18 804
.07 624
.26 428
2
59
.73 591
.92 367
.81 224
.18 776
.07 633
.26 409
1
60
9.73611
9.92 359
9.81 252
0.18 748
0.07 641
0.26 389
0
Cos
Sin
Cot
Tan
Csc
Sec
'
122° C302°)
(237°) 57°
Table 4. Trigonometric Logarithms
229
33° (213°)
(326°) 146°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.73611
9.92 359
9.81 252
0.18 748
0.07 641
0.26 389
60
1
.73 630
.92 351
.81 279
.18721
.07 649
.26 370
59
2
.73 650
.92 343
.81 307
.18 693
.07 657
.26 350
58
3
.73 669
.92 335
.81 335
.18 665
.07 665
.26 331
57
4
.73 689
.92 326
.81 362
.18 638
.07 674
.26311
56
5
9.73 708
9.92 318
9.81 390
0.18610
0.07 682
0.26 292
55
6
.73 727
.92310
.81 418
.18 582
.07 690
.26 273
54
7
.73 747
.92 302
.81 445
.18 555
.07 698
.26 253
53
8
.73 766
.92 293
.81 473
.18 527
.07 707
.26 234
52
9
.73 785
.92 285
.81 500
.18500
.07 715
.26 215
51
10
9.73 805
9.92 277
9.81 528
0.18472
0.07 723
0.26 195 '
50
11
.73 824
.92 269
.81 556
.18444
.07 731
.26 176
49
12
.73 843
.92 260
.81 583
.18417
.07 740
.26 157
48
13
.73 863
.92 252
.81 611
.18 389
.07 748
.26 137
47
14
.73 882
.92 244
.81 638
.18 362
.07 756
.26 118
46
15
9.73 901
9.92 235
9.81 666
0.18 334
0.07 765
0.26 099
45
16
.73 921
.92 227
.81 693
.18307
.07 773
.26 079
44
17
.73 940
.92 219
.81 721
.18 279
.07 781
.26 060
43
18
.73 959
.92211
.81 748
.18 252
.07 789
.26041
42
19
.73 978
.92 202
.81 776
.18 224
.07 798
.26 022
41
20
9.73 997
9.92 194
9.81 803
0.18 197
0.07 806
0.26 003
40
21
.74 017
.92 186
.81 831
.18 169
.07 814
.25 983
39
22
.74 036
.92 177
.81 858
.18 142
.07 823
.25 964
38
23
.74 055
.92 169
.81 886
.18114
.07 831
.25 945
37
24
.74 074
.92 161
.81 913
.18087
.07 839
.25 926
36
25
9.74 093
9.92 152
9.81 941
0.18 059
0.07 848
0.25 907
35
26
.74 113
.92 144
.81 968
.18 032
.07 856
.25 887
34
27
.74 132
.92 136
.81 996
.18004
.07 864
.25 868
33
28
.74 151
.92 127
.82 023
.17 977
.07 873
.25849
32
29
.74 170
.92 119
.82 051
.17 949
.07 881
.25 830
31
30
9.74 189
9.92 111
9.82 078
0.17 922
0.07 889
0.25811
30
31
.74 208
.92 102
.82 106
.17 894
.07 898
.25 792
29
32
.74 227
.92 094
.82 133
.17 867
.07 906
.25 773
28
33
.74 246
.92 086
.82 161
.17 839
.07 914
.25 754
27
34
.74 265
.92 077
.82 188
.17812
.07 923
.25 735
26
35
9.74 284'
9.92 069
9.82 215
0.17 785
0.07 931
0.25 716
25
36
.74 303
.92 060
.82 243
.17 757
.07 940
.25 697
24
37
.74 322
.92 052
.82 270
.17 730
.07 948
.25 678
23
38
.74 341
.92 044
.82 298
.17 702
.07 956
.25 659
22
39
.74 360
.92 035
.82 325
.17 675
.07 965
.25640
21
40
9.74 379
9.92 027
9.82 352
0.17 648
0.07 973
0.25 621
20
41
.74 398
.92 018
.82 380
.17 620
.07 982
.25 602
19
42
.74 417
.92 010
.82 407
.17 593
.07 990
.25 583
18
43
.74 436
.92 002
.82 435
.17 565
.07 998
.25564
17
44
.74 455
.91 993
.82 462
.17 538
.08 007
.25 545
16
45
9.74 474
9.91 985
9.82 489
0.17511
0.08 015
0.25 526
15
46
.74 493
.91 976
.82 517
.17483
.08 024
.25 507
14
47
.74 512
.91 968
.82 544
.17 456
.08 032
.25 488
13
48
.74 531
.91 959
.82 571
.17 429
.08041
.25 469
12
49
.74 549
.91 951
.82 599
.17401
.08049
.25 451
11
50
9.74 568
9.91 942
9.82 626
0.17 374
0.08 058
0.25 432
10
51
.74 587
.91 934
.82 653
.17 347
.08 066
.25 413
9
52
.74 606
.91 925-
.82 681
.17319
.08 075
.25 394
8
53
.74 625
.91 917
.82 708
.17 292
.08 083
.25375
7
54
.74 644
.91 908
.82 735
.17 265
.08 092
.25 356
6
55
9.74 662
9.91 900
9.82 762
0.17 238
0.08 100
0.25 338
5
56
.74 681
.91 891
.82 790
.17210
.08 109
.25 319
4
57
.74 700
.91 883
.82 817
.17183
.08117
.25 300
3
58
.74 719
.91 874
.82844
.17 156
.08 126
.25 281
2
59
.74 737
.91 866
.82 871
.17129
.08 134
.25 263
1
60
9.74 756
9.91 857
9.82 899
0.17 101
0.08 143
0.25 244
0
Cos
Sin
Cot
Tan Csc
Sec
'
123° (303°)
(236°) 56°
230
Table 4. Trigonometric Logarithms
34° (214°)
(325°) 145°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.74 756
9.91 857
9.82 899
0.17 101
0.08 143
0.25 244
60
1
.74 775
.91849
.82 926
.17 074
.08 151
.25 225
59
2
.74 794
.91840
.82 953
.17 047
.08 160
.25 206
58
3
.74 812
.91 832
.82 980
.17 020
.08 168
.25 188
57
4
.74 831
.91 823
.83008
.16 992
.08 177
.25 169
56
5
9.74 850
9.91 815
9.83 035
0.16 965
0.08 185
0.25 150
55
6
.74 868
.91 806
.83 062
.16 938
.08 194
.25 132
54
7
.74 887
.91 798
.83 089
.16911
.08 202
.25 113
53
8
.74 906
.91 789
.83 117
.16 883
.08211
.25 094
52
9
.74 924
.91 781
.83 144
.16 856
.08 219
.25 076
51
10
9.74 943
9.91 772
9.83 171
0.16 829
0.08 228
0.25 057
50
11
.74 961
.91 763
.83 198
.16 802
.08 237
.25 039
49
12
.74 980
.91 755
.83225
.16 775
.08 245
.25 020
48
13
.74 999
.91 746
.83252
.16 748
.08 254
.25 001
47
14
.75 017
.91 738
.83280
•16 720
.08 262
.24 983
46
15
9.75 036
9.91 729
9.83 307
0.16 693
0.08 271
0.24 964
45
16
.75 054
.91 720
.83 334
.16 666
.08 280
.24 946
44
17
.75 073
.91 712
.83361
.16 639
.08 288
.24 927
43
18
.75 091
.91 703
.83 388
.16612
.08 297
.24 909
42
19
.75 110
.91 695
.83415
.16 585
.08 305
.24 890
41
20
9.75 128
9.91 686
9.83 442
0.16 558
0.08 314
0.24 872
40
21
.75 147
.91 677
.83470
.16 530
.08 323
.24 853
39
22
.75 165
.91 669
.83 497
.16 503
.08 331
.24 835
38
23
.75184
.91 660
.83524
.16476
.08 340
.24 816
37
24
.75 202
.91 651
.83551
.16449
.08 349
.24 798
36
25
9.75 221
9.91 643
9.83 578
0.16 422
0.08 357
0.24 779
35
26
.75 239
.91 634
.83 605
.16 395
.08 366
.24 761
34
27
.75 258
.91 625
.83632
.16 368
.08 375
.24 742
33
28
.75 276
.91 617
.83659
.16 341
.08 383
.24 724
32
29
.75 294
.91 608
.83686
.16314
.08 392
.24 706
31
30
9.75 313
9.91 599
9.83 713
0.16 287
0.08 401
0.24 687
30
31
.75 331
.91 591
.83740
.16 260
.08 409
.24 669
29
32
.75 350
.91 582
.83 768
.16 232
.08 418
.24 650
28
33
.75 368
.91 573
.83795
.16 205
.08 427
.24 632
27
34
.75 386
.91 565
.83822
.16 178
.08 435
.24 614
26
35
9.75 405
9.91 556
9.83 849
0.16 151
0.08 444
0.24 595
25
36
.75 423
.91 547
.83 876
.16 124
.08 453
.24 577
24
37
.75 441
.91 538
.83903
.16 097
.08 462
.24 559
23
38
.75 459
.91 530
.83930
.16 070
.08 470
.24 541
22
39
.75 478
.91 521
.83 957
.16043
.08 479
.24 522
21
40
9.75 496
9.91 512
9.83 984
0.16016
0.08 488
0 24 504
20
41
.75 514
.91504
.84011
.15 989
.08 496
.24 486
19
42
.75 533
.91 495
.84038
.15 962
.08 505
.24 467
18
43
.75 551
.91 486
.84065
.15 935
.08 514
.24 449
17
44
.75 569
.91 477
.84092
.15 908
.08 523
.24 431
16
45
9.75 587
9.91 469
9.84 119
0.15 881
0.08 531
0.24 413
15
46
.75 605
.91 460
.84146
.15 854
.08 540
.24 395
14
47
.75 624
.91 451
.84173
.15 827
.08 549
.24 376
13
48
.75642
.91 442
.84200
.15 800
.08 558
.24 358
12
49
.75 660
.91 433
.84227
.15 773
.08 567
.24 340
11
50
9.75 678
9.91 425
9.84 254
0.15 746
0.08 575
0.24 322
10
51
.75 696
.91 416
.84 280
.15 720
.08584
.24 304
9
52
.75 714
.91 407
.84307
.15 693-
.08 593
.24 286
8
53
.75 733
.91 398
.84334
.15 666
.08 602
.24 267
7
54
.75 751
.91 389
.84361
.15 639
.08611
.24 249
6
55
9.75 769
9.91 381
9.84 388
0.15 612
0.08 619
0.24 231
5
56
.75 787
.91 372
.84415
.15 585
.08 628
.24 213
4
57
.75 805
.91 363
.84442
.15 558
.08 637
.24 195
3
58
.75 823
.91 354
.84469
.15 531
.08 646
.24 177
2
59
.75841
.91 345
.84496
.15 504
.08 665
.24 159
1
60
9.75 859
9.91 336
9.84 523
0.15 477
0.08 664
0.24 141
0
Cos
Sin
Cot
Tan
Csc
Sec
'
124° (304°)
(235°) 55°
Table 4. Trigonometric Logarithms
231
35° (215°)
(324°) 144°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.75 859
9.91 336
9.84 523
0.15477
0.08 664
0.24 141
60
1
.75 877
.91 328
.84550
.15 450
.08 672
.24 123
59
2
.75 895
.91 319
.84576
.15 424
.08 681
.24 105
58
3
.75 913
.91 310
.84603
.15 397
.08 690
.24087
57
4
.75 931
.91 301
.84630
.15 370
.08 699
.24 069
56
5
9.75 949
9.91 292
9.84 657
0.15 343
0.08 708
0.24 051
55
6
.75 967
.91 283
.84684
.15316
.08 717
.24 033
54
7
.75985
.91 274
.84711
.15 289
.08 726
.24 015
53
8
.76003
.91 266
.84738
.15 262
.08 734
.23 997
52
9
.76 021
.91 257
.84764
.15 236
.08 743
.23 979
51
10
9.76 039
9.91 248
9.84 791
0.15209
0.08 752
0.23 961
50
11
.76 057
.91 239
.84818
.15 182
.08 761
.23 943
49
12
.76 075
.91 230
.84845
.15 155
.08 770
.23 925
48
13
.76 093
.91 221
.84872
.15 128
.08 779
.23 907
47
14
.76111
.91 212
.84899
.15 101
.08 788
.23 889
46
15
9.76 129
9.91 203
9.84 925
0.15 075
0.08 797
0.23 871
45
16
.76 146
.91 194
.84952
.15 048
.08 806
.23 854
44
17
.76164
.91 185
.84979
.15 021
.08 815
.23 836
43
18
.76 182
.91 176
.85 006
.14 994
.08 824
.23 818
42
19
.76 200
.91 167
.85033
.14 967
.08 833
.23800
41
20
9.76 218
9.91 158
9.85 059
0.14 941
0.08 842
0.23 782
40
21
.76 236
.91 149
.85086
.14914
.08851
.23764
39
22
.76 253
.91 141
.85113
.14 887
.08 859
.23 747
38
23
.76 271
.91 132
.85 140
.14 860
.08 868
.23 729
37
24
.76 289
.91 123
.85 166
.14 834
.08 877
.23 711
36
25
9.76 307
9.91 114
9.85 193
0.14 807
0.08 886
0.23 693
35
26
.76 324
.91 105
.85 220
.14 780
.08 895
.23 676
34
27
.76 342
.91 096
.85 247
.14 753
.08 904
.23 658
33
28
.76 360
.91 087
.85 273
.14 727
.08 913
.23 640
32
29
.76 378
.91 078
.85 300
.14 700
.08 922
.23 622
31
30
9.76 395
9.91 069
9.85 327
0.14 673
0.08 931
0.23 605
30
31
.76 413
.91 060
.85 354
.14646
.08 940
.23 587
29
32
.76 431
.91 051
.85 380
.14 620
.08 949
.23 569
28
33
.76 448
.91 042
.85 407
.14 593
.08 958
.23 552
27
34
.76 466
.91 033
.85 434
.14 566
.08 967
.23 534
26
35
9.76 484
9.91 023
9.85 460
0.14 540
0.08 977
0.23 516
25
36
.76 501
.91 014
.85 487
.14513
.08 986
.23 499
24
37
.76 519
.91 005
.85 514
.14 486
.08 995
.23 481
23
38
.76 537
.90 996
.85 540
.14 460
.09004
.23 463
22
39
.76 554
.90 987
.85567
.14433
.09 013
.23446
21
40
9.76 572
9.90 978
9.85 594
0.14 406
0.09 022
0.23 428
20
41
.76 590
.90 969
.85620
.14 380
.09 031
.23 410
19
42
.76 607
.90 960
.85647
.14 353
.09040
.23 393
18
43
.76 625
.90 951
.85674
.14 326
.09049
.23 375
17
44
.76642
.90 942
.85700
.14 300
.09 058
.23 358
16
45
9.76 660
9.90 933
9.85 727
0.14 273
0.09 067
0.23 340
15
46
.76 677
.90 924
.85 754
.14 246
.09 076
.23 323
14
47
.76 695
.90 915
.85780
. .14 220
.09 085
.23 305
13
48
.76 712
.90 906
.85807
.14 193
.09 094
.23288
12
49
.76 730
.90 896
.85834
.14 166
.09104
.23 270
11
50
9.76 747
9.90 887
9.85 860
0.14 140
0.09 113
0.23 253
10
51
.76 765
.90 878
.85887
.14 113
.09 122
.23 235
9
52
.76 782
.90 869
.85913
.14 087
.09 131
.23 218
8
53
.76800
.90860
.85940
.14 060
.09 140
.23 200
7
54
.76 817
.90851
.85967
.14 033
.09 149
.23 183
6
55
9.76 835
9.90 842
9.85 993
0.14 007
0.09 158
0.23 165
5
56
.76852
.90 832
.86 020
.13 980
.09 168
.23 148
4
57
.76 870
.90 823
.86046
.13 954
.09 177
.23 130
3
58
.76 887
.90 814
.86 073
.13 927
.09186
.23 113
2
59
.76904
.90 805
.86100
.13900
.09 195
.23 096
1
60
9.76 922
9.90 796
9.86 126
0.13 874
0.09 204
0.23 078
0
Cos
Sin
Cot
Tan
Csc
Sec
'
125° (305°)
(234°) 54°
232
Table 4. Trigonometric Logarithms
36° (216°)
(323°) 143°
'
Sin
Cos
Tan
Cot
Sec
('so
0
9.76 922
9.90 796
9.86 126
0.13 874
0.09 204
0.23 078
60
1
.76 939
.90 787
.86 153
.13847
.09 213
.23 061
59
2
.76 957
.90 777
.86 179
.13 821
.09 223
.23 043
58
3
.76 974
.90 768
.86 206
.13 794
.09 232
.23 026
57
4
.76 991
.90 759
.86 232
.13 768
.09 241
.23 009
56
5
9.77 009
9.90 750
9.86 259
0.13 741
0.09 250
0.22 991
55
6
.77 026
.90 741
.86 285
.13715
.09 259
.22 974
54
7
.77 043
.90 731
.86 312
.13 688
.09 269
.22 957
53
8
.77 061
.90 722
.86 338
.13 662
.09 278
.22 939
52
9
.77 078
.90 713
.86 365
.13 635
.09 287
.22 922
51
10
9.77 095
9.90 704
9.86 392
0.13 608
0.09 296
0.22 905
50
11
.77112
.90 694
.86 418
.13 582
.09 306
.22 888
49
12
.77 130
.90 685
.86 445
.13 555
.09 315
.22 870
48
13
.77 147
.90 676
.86 471
.13 529
.09 324
.22 853
47
14
.77 164
.90 667
.86 498
.13 502
.09 333
.22 836
46
15
9.77 181
9.90 657
9.86 524
0.13 476
0.09 343
0.22 819
45
16
.77 199
.90648
.86 551
.13 449
.09 352
.22 801
44
17
.77 216
.90 639
.86 577
.13 423
.09 361
.22784
43
18
.77 233
.90 630
.86 603
.13 397
.09 370
.22 767
42
19
.77 250
.90 620
.86 630
.13 370
.09 380
.22 750
41
20
9.77 268
9.90611
9.86 656
0.13 344
0.09 389
0.22 732
40
21
.77 285
.90 602
.86 683
.13317
.09 398
.22 715
39
22
.77 302
.90 592
.86 709
.13 291
.09 408
.22 698
38
23
.77 319
.90 583
.86 736
.13 264
.09 417
.22 681
37
24
.77 336
.90 574
.86762
.13 238
.09 426
.22 664
36
25
9.77 353
9.90 565
9.86 789
0.13211
0.09 435
0.22 647
35
26
.77 370
.90 555
.86 815
.13 185
.09 445
.22 630
34
27
.77 387
.90 546
.86842
.13 158
.09 454
.22 613
33
28
.77 405
.90 537
.86 868
.13 132
.09 463
.22 595
32
29
.77 422
.90 527
.86 894
•13 106
.09 473
.22 578
31
30
9.77 439
9.90 518
9.86 921
0.13 079
0.09 482
0.22 561
30
' 31
.77 456
.90 509
.86 947
.13 053
.09 491
.22 544
29
32
.77 473
.90 499
.86 974
.13 026
.09 501
.22 527
28
33
.77 490
.90 490
.87000
.13 000
.09 510
.22 510
27
34
.77 507
.90 480
.87 027
.12 973
.09 520
.22 493
26
35
9.77 524
9.90 471
9.87 053
0.12 947
0.09 529
0.22 476
25
36
.77 541
.90 462
.87 079
.12 921
.09 538
.22 459
24
37
.77 558
.90 452
.87 106
.12 894
.09 548
.22 442
23
38
.77 575
.90 443
.87 132
.12 868
.09 557
.22 425
22
39
.77 592
.90 434
.87 158
.12842
.09 566
.22 408
21
40
9.77 609
9.90 424
9.87 185
0.12815
0.09 576
0.22 391
20
41
.77 626
.90 415
.87211
.12 789
.09 585
.22 374
19
42
.77 643
.90 405
.87 238
.12 762
.09 595
.22 357
18
43
.77 660
.90 396
.87 264
.12 736
.09 604
.22 340
17
44
.77 677
.90 386
.87 290
.12710
.09 614
.22 323
16
45
9.77 694
9.90 377
9.87 317
0.12683
0.09 623
0.22 306
15
46
.77711
.90 368
.87 343
.12 657
.09 632
.22 289
14
47
.77 728
.90 358
.87 369
.12 631
.09 642
22272
13
48
.77 744
.90 349
.87 396
.12604
.09 651
.22 256
12
49
.77 761
.90 339
.87 422
.12 578
.09 661
.22 239
11
50
9.77 778
9.90 330
9.87 448
0.12 552
0.09 670
0.22 222
10
51
.77 795
.90 320
.87 475
.12 525
.09 680
.22 205
9
52
.77 812
.90311
.87 501
.12 499
.09 689
.22 188
8
53
.77 829
.90 301
.87 527
.12 473
.09 699
.22 171
7
54
.77846
.90 292
.87 554
.12 446
.09 708
.22 154
6
55
9.77 862
9.90 282
9.87 580
0.12 420
0.09 718
0.22 138
5
56
.77 879
.90 273
.87 606
.12 394
.09 727
.22 121
4
57
.77 896
.90 263
.87 633
.12 367
'.09737
.22 104
3
58
.77 913
.90 254
.87 659
.12 341
.09 746
.22 087
2
59
.77 930
.90 244
.87 685
.12315
.09756
.22 070
1
60
9.77 946
9.90 235
9.87711
0.12 289
0.09 765
0.22 054
0
Cos
Sin
Cot
Tan
Csc
Sec
'
126° (306°)
(233°) 53°
Table 4. Trigonometric Logarithms
233
37° (217°)
(322°) 142 c
'
Sin
Cos
Tan
Cot
Sec Csc
0
9.77 946
9.90 235
9.87711
0.12289
0.09 765
0.22 054
60
1
.77 963
.90 225
.87 738
.12 262
.09 775
.22 037
59
2
.77 980
.90 216
.87 764
.12 236
.09 784
.22 020
58
3
.77 997
.90 206
.87 790
.12210
.09794
.22 003
57
4
.78 013
.90 197
.87 817
.12 183
.09803
.21 987
56
5
9.78 030
9.90 187
9.87 843
0.12 157
0.09 813
0.21 970
55
6
.78047
.90 178
.87 869
.12 131
.09 822
.21 953
54
7
.78063
.90 168
.87 895
.12 105
.09 832
.21 937
53
8
.78 080
.90 159
.87 922
.12 078
.09841
.21 920
52
9
.78 097
.90 149
.87 948
.12 052
.09 851
.21 903
51
10
9.78 113
9.90 139
9.87 974
0.12 026
0.09 861
0.21 887
50
11
.78 130
.90 130
.88 000
.12 000
.09 870
.21 870
49
12
.78 147
.90 120
.88 027
.11 973
.09 880
.21 853
48
13
.78 163
.90111
.88 053
.11 947
.09 889
.21 837
47
14
.78 180
.90 101
.88 079
.11 921
.09 899
.21 820
46
15
9.78 197
9.90 091
9.88 105
0.11 895
0.09 909
0.21 803
45
16
.78 213
.90 082
.88131
.11 869
.09 918
.21 787
44
17
.78 230
.90 072
.88 158
.11 842
.09 928
.21 770
43
18
.78 246
.90 063
.88184
.11 816
.09 937
.21 754
42
19
.78 263
.90 053
.88 210
.11 790
.09 947
.21 737
41
20
9.78 280
9.90 043
9.88 236
0.11 764
0.09 957
0.21 720
40
21
.78 296
.90 034
.88 262 .
.11 738
.09 966
.21 704
39
22
.78313
.90 024
.88289
.11 711
.09 976
.21 687
38
23
.78 329
.90 014
.88315
.11 685
.09 986
.21 671
37
24
.78 346
.90 005
.88341
.11659
.09 995
.21 654
36
25
9.78 362
9.89 995
9.88 367
0.11633
0.10 005
0.21 638
35
26
.78 379
.89 985
.88 393
.11607
.10015
.21 621
34
27
.78 395
.89 976
.88420
.11580
.10 024
.21 605
33
28
.78412
.89 966
.88 446
.11 554
.10 034
.21588
32
29
.78 428
.89 956
.88472
.11528
.10 044
.21 572
31
30
9.78 445
9.89 947
9.88 498
0.11 502
0.10 053
0.21 555
30
31
.78 461
.89 937
.88524
.11476
.10 063
.21 539
29
32
.78 478
.89 927
.88 550
.11 450
.10 073
.21 522
28
33
.78 494
.89 918
.88 577
.11423
.10 082
.21 506
27
34
.78510
.89 908
.88 603
.11 397
.10 092
.21 490
26
35
9.78 527
9.89 898
9.88 629
0.11 371
0.10 102
0.21 473
25
36
.78 543
.89 888
.88 655
.11 345
.10112
.21 457
24
37
.78 560
.89 879
.88 681
.11319
.10121
.21 440
23
38
.78 576
.89 869
.88 707
.11 293
.10 131
.21 424
22
39
.78 592
.89859
.88733
.11 267
.10 141
.21 408
21
40
9.78 609
9.89 849
9.88 759
0.11 241
0.10 151
0.21 391
20
41
.78 625
.89840
.88 786
.11 214
.10 160
.21 375
19
42
.78 642
.89830
.88812
.11 188
.10 170
.21 358
18
43
.78 658
.89 820
.88838
.11 162
.10 180
.21 342
17
44
.78 674
.89 810
.88864
.11 136
.10 190
.21 326
16
45
9.78 691
9.89 801
9.88 890
0.11 110
0.10 199
0.21 309
15
46
.78 707
.89 791
.88916
.11 084
.10 209
.21 293
14
47
.78 723
.89 781
.88 942
.11058
.10219
.21 277
13
48
.78 739
.89 771
.88968
.11032
.10 229
.21 261
12
49
.78 756
.89 761
.88 994
.11 006
.10 239
.21 244
11
50
9.78 772
9.89 752
9.89 020
0.10980
0.10 248
0.21 228
10
51
.78 788
.89 742
.89046
.10 954
.10258
.21 212
9
52
.78805
.89 732
.89 073
.10 927
.10 268
.21 195
8
53
.78 821
.89 722
.89 099
.10901
.10 278
.21 179
7
54
.78837
.89 712
.89 125
.10 875
.10 288
.21 163
6
55
9.78 853
9.89 702
9.89 151
0.10 849
0.10 298
0.21 147
5
56
.78 869
.89 693
.89 177
.10 823
.10 307
.21 131
4
57
.78886
.89683
.89 203
.10 797
.10317
.21 114
3
58
.78 902
.89 673
.89 229
.10771
.10 327
.21 098
2
59
.78 918
.89 663
.89 255
.10745
.10 337
.21 082
1
60
9.78 934
9.89 653
9.89 281
0.10719
0.10 347
0.21 066
0
Cos
Sin
Cot
Tan
Csc
Sec
'
127° (307°)
(232°) 52°
234
Table 4. Trigonometric Logarithms
38° (218°)
(321°) 141°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.78 934
9.89 653
9.89 281
0.10719
0.10 347
0.21 066
60
1
.78 950
.89 643
.89 307
.10 693
.10 357
.21 050
59
2
.78 967
.89 633
.89 333
.10 667
.10367
.21 033
58
3
.78 983
.89 624
.89 359
.10641
.10 376
.21 017
57
4
.78 999
.89 614
.89 385
.10615
.10 386
.21 001
56
5
9.79 015
9.89 604
9.89411
0.10 589
0.10 396
0.20 985
55
6
.79 031
.89 594
.89 437
.10 563
.10 406
.20 969
54
7
.79047
.89 584
.89 463
.10 537
.10416
.20 953
53
8
.79 063
.89 574
.89 489
.10511
.10426
.20 937
52
9
.79 079
.89 564
.89 515
.10485
.10 436
.20 921
51
10
9.79 095
9.89 554
9.89 541
0.10459
0.10446
0.20 905
50
11
.79111
.89 544
.89 567
.10433
.10 456
.20 889
49
12
.79 128
.89 534
.89 593
.10407
.10466
.20 872
48
13
.79 144
.89 524
.89 619
.10381
.10476
.20 856
47
14
.79 160
.89 514
.89645
.10 355
.10 486
.20840
46
15
9.79 176
9.89 504
9.89 671
0.10 329
0.10496
0.20 824
45
16
.79 192
.89 495
.89 697
.10 303
.10 505
.20 808
44
17
.79 208
.89 485
.89 723
.10 277
.10515
.20 792
43
18
.79 224
.89 475
.89 749
.10251
.10 525
.20 776
42
19
.79 240
.89 465
.89 775
.10 225
.10 535
.20 760
41
20
9.79 256
9.89 455
9.89 801
0.10 199
0.10545
0.20 744
40
21
.79 272
.89 445
.89 827
.10 173
.10 555
.20 728
39
22
.79 288
.89 435
.89 853
.10 147
.10 565
.20712
38
23
.79 304
.89 425
.89 879
.10 121
.10 575
.20 696
37
24
.79 319
.89 415
.89 905
.10 095
.10 585
.20 681
36
25
9.79 335
9.89 405
9.89 931
0.10 069
0.10 595
0.20 665
35
26
.79 351
.89 395
.89 957
.10043
.10 605
.20 649
34
27
.79 367
.89 385
.89 983
.10017
.10615
.20 633
33
28
.79 383
.89 375
.90 009
.09 991
.10 625
.20 617
32
29
.79 399
.89364
.90 035
.09 965
.10 636
.20 601
31
30
9.79 415
9.89 354
9.90 061
0.09 939
0.10 646
0.20 585
30
31
.79 431
.89 344
.90 086
.09 914
.10 656
.20 569
29
32
.79447
.89 334
.90 112
.09 888
.10 666
.20 553
28
33
.79 463
.89 324
.90 138
.09 862
.10 676
.20 537
27
34
.79 478
.89 314
.90164
.09 836
.10 686
.20 522
26
35
9.79 494
9.89 304
9.90 190
0.09 810
0.10 696
0.20 506
25
36
.79 510
.89 294
.90 216
.09784
.10 706
.20490
24
37
.79 526
.89284
.90 242
.09 758
.10716
.20 474
23
38
.79 542
.89 274
.90 268
.09 732
.10 726
.20 458
22
39
.79 558
.89 264
.90 294
.09 706
.10 736
.20 442
21
40
9.79 573
9.89 254
9.90 320
0.09 680
O'lO 746
0.20 427
20
41
.79 589
.89 244
.90 346
.09 654
.10 756
.20411
19
42
.79 605
.89 233
.90 371
.09 629
.10 767
.20 395
18
43
.79 621
.89 223
.90 397
.09 603
.10 777
.20 379
17
44
.79 636
.89 213
.90 423
.09 577
.10 787
.20 364
16
45
9.79 652
9.89 203
9.90 449
0.09 551
0.10797
0.20 348
15
46
.79 668
.89 193
.90 475
.09 525
.10 807
.20 332
14
47
.79684
.89 183
.90 501
.09 499
.10817
.20316
13
48
.79 699
.89 173
.90 527
.09 473
.10 827
.20 301
12
49
.79 715
.89 162
.90 553
.09 447
.10 838
.20 285
11
50
9.79 731
9.89 152
9.90 578
0.09 422
0.10 848
0.20 269
10
51
.79 746
.89 142
.90604
.09 396
.10 858
.20 254
9
52
.79 762
.89 132
.90 630
.09 370
.10 868
.20 238
8
53
.79 778
.89 122
.90 656
.09 344
.10 878
.20 222
7
54
.79 793
.89 112
.90 682
.09 318
.10 888
.20 207
6
55
9.79 809
9.89 101
9.90 708
0.09 292
0.10 899
0.20 191
5
56
.79 825
.89 091
.90 734
.09 266
.10 909
.20 175
4
57
.79840
.89 081
.90 759
.09 241
.10919
.20 160
3
58
.79 856
.89 071
.90 785
.09 215
.10 929
.20 144
2
59
.79 872
.89 060
.90811
.09 189
.10 940
.20 128
1
60
9.79 887
9.89 050
9.90 837
0.09 163
0.10 950
0.20 113
0
Cos
Sin
Cot
Tan
Csc
Sec
'
128° (308°)
(231°) 51°
Table 4. Trigonometric Logarithms
235
39° (219°)
(320°) 140°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.79 887
9.89 050
9.90 837
0.09 163
0.10950
0.20 113
60
1
.79 903
.89 040
.90 863
.09 137
.10 960
.20 097
59
2
.79 918
.89 030
.90 889
.09111
.10970
.20 082
58
3
.79 934
.89 020
.90 914
.09 086
.10 980
.20 066
57
4
.79 950
.89 009
.90 940
.09 060
.10 991
.20 050
56
5
9.79 965
9.88 999
9.90 966
0.09 034
0.11 001
0.20 035
55
6
.79 981
.88 989
.90 992
.09 008
.11011
.20 019
54
7
.79 996
.88 978
.91 018
.08 982
.11 022
.20 004
53
8
.80 012
.88968
.91 043
.08 957
.11 032
.19 988
52
9
.80 027
.88958
.91 069
.08 931
.11 042
.19 973
51
10
9.80 043
9.88 948
9.91 095
0.08 905
0.11 052
0.19 957
50
11
.80058
.88 937
.91 121
.08 879
.11 063
.19 942
49
12
.80 074
.88927
.91 147
.08 853
.11 073
.19 926
48
13
.80 089
.88 917
.91 172
.08 828
.11083
.19911
47
14
.80 105
.88906
.91 198
.08 802
.11 094
.19 895
46
15
9.80 120
9.88 896
9.91 224
0.08 776
0.11 104
0.19 880
45
16
.80136
.88886
.91 250
.08 750
.11 114
.19864
44
17
.80 151
.88 875
.91 276
.08 724
.11 125
.19 849
43
18
.80 166
.88865
.91 301
.08 699
.11 135
.19834
42
19
.80 182
.88855
.91 327
.08 673
.11 145
.19818
41
20
9.80 197
9.88 844
9.91 353
0.08 647
0.11 156
0.19 803
40
21
.80 213
.88 834
.91 379
.08 621
.11 166
.19 787
39
22
.80 228
.88824
.91 404
.08 596
.11 176
.19 772
38
23
.80 244
.88 813
.91 430
.08 570
.11 187
.19 756
37
24
.80 259
.88803
.91 456
.08 544
.11 197
.19 741
36
25
9.80 274
9.88 793
9.91 482
0.08 518
0.11 207
0.19 726
35
26
.80 290
.88 782
.91 507
.08 493
.11 218
.19710
34
27
.80 305
.88 772
.91 533
.08 467
.11 228
.19 695
33
28
.80 320
.88 761
.91 559
.08 441
.11 239
.19 680
32
29
.80 336
.88 751
.91 585
.08 415
.11 249
.19 664
31
30
9.80 351
9.88 741
9.91 610
0.08 390
0.11 259
0.19 649
30
31
.80 366
.88730
.91 636
.08364
.11 270
.19 634
29
32
.80 382
.88 720
.91 662
.08 338
.11 280
.19618
28
33
.80 397
.88 709
.91 688
.08 312
.11291
.19 603
27
34
.80 412
.88 699
.91 713
.08 287
.11 301
.19 588
26
35
9.80 428
9.88 688
9.91 739
0.08 261
0.11312
0.19 572
25
36
.80 443
.88 678
.91 765
.08 235
.11 322
.19 557
24
37
.80 458
.88 668
.91 791
.08 209
.11332
.19 542
23
38
.80 473
.88 657
.91 816
.08184
.11343
.19 527
22
39
.80 489
.88 647
.91842
.08 158
.11 353
.19511
21
40
9.80 504
9.88 636
9.91 868
0.08 132
0.11 364
0.19 496
20
41
.80 519
.88 626
.91 893
.08 107
.11 374
.19481
19
42
.80 534
.88615
.91 919
.08 081
.11385
.19466
18
43
.80 550
.88605
.91 945
.08 055
.11 395
.19450
17
44
.80 565
.88594
.91 971
.08 029
.11406
.19 435
16
45
9.80 580
9.88 584
9.91 996
0.08 004
0.11416
0.19 420
15
46
.80 595
.88 573
.92 022
.07 978
.11427
.19 405
14
47
.80 610
.88 563
.92048
.07 952
.11437
.19 390
13
48
.80 625
.88 552
.92 073
.07 927
.11448
.19 375
12
49
.80641
.88 542
.92 099
.07 901
.11458
.19 359
11
50
9.80 656
9.88 531
9.92 125
0.07 875
0.11 469
0.19344
10
51
.80671
.88 521
.92 150
.07 850
.11479
.19 329
9
52
.80686
.88 510
.92 176
.07 824
.11 490
.19314
8
53
.80701
.88 499
.92 202
.07 798
.11 501
.19 299
7
54
.80716
.88 489
.92 227
.07 773
.11511
.19284
6
55
9.80 731
9.88 478
9.92 253
0.07 747
0.11 522
0.19 269
5
56
.80746
.88468
.92 279
.07 721
.11 532
.19 254
4
57
.80762
.88 457
.92304
.07 696
.11 543
.19 238
3
58
.80 777
.88447
.92 330
.07 670
.11 553
.19 223
2
59
.80 792
.88 436
.92 356
.07 644
.11564
.19 208
1
60
9. so s()7
9.88 425
9.92 381
0.07 619
0.11 575
0.19 193
0
Cos
Sin
Cot
Tan
Csc
Sec '
129° (309°)
(230°) 50°
236
Table 4. Trigonometric Logarithms
40° (220°)
(319°) 139°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.80 807
9.88 425
9.92 381
0.07 619
0.11575
0.19 193
60
1
.80 822
.88415
.92 407
.07 593
.11585
.19 178
59
2
.80 837
.88 404
.92 433
.07 567
.11 596
.19 163
58
3
.80 852
.88 394
.92 458
.07 542
.11 606
.19 148
57
4
.80 867
.88383
.92 484
.07 516
.11 617
.19 133
56
5
9.80 882
9.88 372
9.92 510
0.07 490
0.11 628
0.19 118
55
6
.80 897
.88 362
.92 535
.07 465
.11 638
.19 103
54
7
.80 912
.88 351
.92 561
.07 439
.11 649
.19 088
53
8
.80 927
.88340
.92 587
.07 413
.11 660
.19 073
52
9
.80 942
.88 330
.92 612
.07 388
.11 670
.19 058
51
10
9.80 957
9.88 319
9.92 638
0.07 362
0.11 681
0.19 043
50
11
.80 972
.88308
.92 663
.07 337
.11692
.19 028
49
12
.80 987
.88 298
.92 689
.07311
.11 702
.19013
48
13
.81 002
.88 287
.92 715
.07 285
.11713
.18 998
47
14
.81 017
.88276
.92 740
•07 260
.11 724
.18 983
46
15
9.81 032
9.88 266
9.92 766
0.07 234
0.11 734
0.18 968
45
16
.81 047
.88 255
.92 792
.07 208
.11745
.18 953
44
17
.81 061
.88 244
.92 817
.07 183
.11 756
.18 939
43
18
.81076
.88 234
.92 843
.07 157
.11 766
.18 924
42
19
.81 091
.88223
.92 868
•07 132
.11 777
.18 909
41
20
9.81 106
9.88 212 •
9.92 894
0.07 106
0.11 788
0.18 894
40
21
.81 121
.88 201
.92 920
.07 080
.11 799
.18 879
39
22
.81 136
.88 191
.92 945
.07 055
.11 809
.18 864
38
23
.81 151
.88 180
.92 971
.07 029
.11 820
.18849
37
24
.81 166
.88 169
.92996
.07 004
.11 831
.18 834
36
25
9.81 180
9.88 158
9.93 022
0.06 978
0.11 842
0.18 820
35
26
.81 195
.88 148
.93 048
.06 952
.11 852
.18 805
34
27
.81 210
.88 137
.93 073
.06 927
.11 863
.18 790
33
28
.81 225
.88 126
.93 099
.06 901
.11 874
.18 775
32
29
.81 240
.88 115
.93 124
.06 876
.11 885
.18 760
31
30
9.81 254
9.88 105
9.93 150
0.06 850
0.11 895
0.18 746
30
31
.81 269
.88 094
.93 175
.06 825
.11 906
.18731
29
32
.81 284
.88 083
.93 201
.06 799
.11 917
.18716
28
33
.81 299
.88 072
.93 227
.06 773
.11 928
.18701
27
34
.81 314
.88 061
.93 252
.06 748
.11 939
.18 686
26
35
9.81 328
9.88 051
9.93 278
0.06 722
0.11 949
0.18 672
25
36
.81 343
.88040
.93 303
.06 697
.11 960
.18 657
24
37
.81 358
.88029
.93 329
.06 671
.11971
.18 642
23
38
.81 372
.88 018
.93 354
.06 646
.11 982
.18628
22
39
.81 387
.88 007
.93 380
.06 620
.11 993
.18613
21
40
9.81 402
9.87 996
9.93 406
0.06 594
0.12 004
0.18598
20
41
.81 417
.87 985
.93 431
.06 569
.12015
.18 583
19
42
.81 431
.87 975
.93 457
.06 543
.12 025
.18 569
18
43
.81 446
.87 964
.92 482
.06 518
.12 036
.18554
17
44
.81 461
.87 953
.93 508
.06 492
.12 047
.18539
16
45
9.81 475
9.87 942
9.93 533
0.06 467
0.12 058
0.18525
15
46
.81 490
.87 931
.93 559
.06 441
.12 069
.18510
14
47
.81 505
.87 920
.93584
.06 416
.12 080
.18 495
13
48
.81 519
.87 909
.93 610
.06 390
.12 091
.18 481
12
49
.81 534
.87 898
.93 636
.06 364
.12 102
.18 466
11
50
9.81 549
9.87 887
9.93 661
0.06 339
0.12 113
0.18451
10
51
.81 563
.87 877
.93 687
.06 313
.12 123
.18437
9
52
.81 578
.87 866
.93 712
.06 288
.12 134
.18422
8
53
.81 592
.87 855
.93 738
.06 262
.12 145
.18 408
7
54
.81 607
.87844
.93 763
.06 237
.12 156
.18393
6
55
9.81 622
9.87 833
9.93 789
0.06211
0.12 167
0.18 378
5
56
.81 636
.87 822
.93 814
.06 186
.12 178
.18 364
4
57
.81 651
.87811
.93 840
.06 160
.12 189
.18 349
3
58
.81 665
.87 800
.93 865
.06 135
.12 200
.18 335
2
59
.81 680
.87 789
.93 891
.06 109
.12211
.18 320
1
60
9.81 694
9.87 778
9.93 916
0.06 084
0.12 222
.18 306
0
Cos
Sin
Cot
Tan
Csc
Sec
'
130° (310°)
(229°) 49°
Table 4. Trigonometric Logarithms
237
41° (221°)
(318°) 138°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.81 694
9.87 778
9.93 916
0.06 084
0.12222
0.18306
60
1
.81 709
.87 767
.93 942
.06 058
.12 233
.18291
59
2
.81 723
.87 756
.93 967
.06 033
.12 244
.18277
58
3
.81 738
.87 745
.93 993
.06 007
.12255
.18262
57
4
.81 752
.87 734
.94 018
.05 982
.12 266
.18248
56
5
9.81 767
9.87 723
9.94 044
0.05 956
0.12 277
0.18233
55
6
.81 781
.87 712
.94 069
.05 931
.12 288
.18219
54
7
.81 796
.87 701
.94 095
.05 905
.12 299
.18204
53
8
.81 810
.87 690
.94 120
.05880
.12310
.18 190
52
9
.81 825
.87 679
.94 146
.05 854
.12 321
.18 175
51
10
9.81 839
9.87 668
9.94 171
0.05 829
0.12332
0.18 161
50
11
.81854
.87 657
.94 197
.05 803
.12 343
.18 146
49
12
.81 868
.87646
.94 222
-.05 778
.12 354
.18 132
48
13
.81 882
.87 635
.94248
.05 752
.12 365
.18 118
47
14
.81 897
.87 624
.94 273
.05 727
.12 376
.18 103
46
15
9.81 911
9.87 613
9.94 299
0.05 701
0.12 387
0.18 089
45
16
.81 926
.87 601
.94324
.05 676
.12 399
.18074
44
17
.81 940
.87 590
.94 350
.05 650
.12410
.18 060
43
18
.81 955
.87 579
.94 375
.05 625
.12421
.18 045
42
19
.81 969
.87 568
.94401
.05 599
.12432
.18031
41
20
9.81 983
9.87 557
9.94 426
0.05 574
0.12 443
0.18017
40
21
.81 998
.87 546
.94 452
.05 548
.12 454
.18 002
39
22
.82 012
.87 535
.94 477
.05 523
.12 465
.17 988
38
23
.82 026
.87 524
.94 503
.05 497
.12 476
.17 974
37
24
.82041
.87 513
.94 528
.05 472
.12487
.17 959
36
25
9.82 055
9.87 501
9.94 554
0.05 446
0.12 499
0.17 945
35
26
.82 069
.87 490
.94 579
.05 421
.12510
.17 931
34
27
.82084
.87 479
.94604
.05 396
.12521
.17916
33
28
.82 098
.87 468
.94 630
.05 370
.12 532
.17 902
32
29
.82112
.87 457
.94 655
.05 345
.12 543
.17 888
31
30
9.82 126
9.87 446
9.94 681
0.05 319
0.12 554
0.17 874
30
31
.82 141
.87 434
.94706
.05 294
.12 566
.17859
29
32
.82 155
.87 423
.94 732
.05 268
.12577
.17845
28
33
.82 169
.87 412
.94 757
.05 243
.12 588
.17831
27
34
.82184
.87 401
.94783
•05 217
.12 599
.17816
26
35
9.82 198
9.87 390
9.94 808
0.05 192
0.12610
0.17 802
25
36
.82 212
.87 378
.94 834
.05 166
.12 622
.17 788
24
37
.82 226
.87 367
.94 859
.05 141
.12 633
.17 774
23
38
.82 240
.87 356
.94884
.05 116
.12644
.17 760
22
39
.82 255
.87 345
.94 910
.05 090
.12 655
.17 745
21
40
9.82 269
9.87 334
9.94 935
0.05 065
0.12 666
0.17731
20
41
.82 283
.87 322
.94 961
.05 039
.12 678
.17717
19
42
.82 297
.87311
.94 986
.05 014
.12 689
.17 703
18
43
.82311
.87 300
.95 012
.04988
.12 700
.17 689
17
44
.82 326
.87 288
.95 037
.04963
.12712
.17 674
16
45
9.82 340
9.87 277
9.95 062
0.04 938
0.12 723
0.17 660
15
46
.82 354
.87 266
.95 088
.04 912
.12 734
.17646
14
47
.82 368
.87 255
.95113
.04887
.12 745
.17 632
13
48
•82 382
.87 243
.95 139
.04861
.12 757
.17618
12
49
.82 396
.87 232
.95 164
.04836
.12 768
.17604
11
50
9.82 410
9.87 221
9.95 190
0.04 810
0.12 779
0.17590
10
51
.82 424
.87 209
.95 215
.04785
.12 791
.17 576
9
52
.82 439
.87 198
.95 240
.04760
.12 802
.17561
8
53
.82 453
.87 187
.95 266
.04734
.12813
.17 547
7
54
.82 467
.87 175
.95 291
.04709
.12 825
.17533
6
55
9.82 481
9.87 164
9.95 317
0.04 683
0.12 836
0.17519
5
56
.82 495
.87 153
.95 342
.04658
.12847
.17 505
4
57
.82 509
.87 141
.95 368
.04632
.12 859
.17491
3
58
.82 523
.87 130
.95 393
.04607
.12 870
.17 477
2
59
.82 537
.87119
.95418
.04582
.12881
.17 463
1
60
9.82 551
9.87 107
9.95 444
0.04 556
0.12 893
0.17 449
0
Cos
Sin
Cot
Tail
Csc
Sec
'
131° (311°)
(228°) 48°
238
Table 4. Trigonometric Logarithms
42° (222°)
(317°) 137°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.82 551
9.87 107
9.95 444
0.04 556
0.12 893
0.17 449
60
1
.82 565
.87 096
.95 469
.04 531
.12904
.17 435
59
2
.82 579
.87 085
.95 495
.04 505
.12915
.17 421
58
3
.82 593
.87 073
.95 520
.04 480
.12 927
.17 407
57
4
.82 607
.87 062
.95 545
.04455
.12 938
.17 393
56
5
9.82 621
9.87 050
9.95 571
0.04 429
0.12 950
0.17 379
55
6
.82 635
.87 039
.95 596
.04404
.12 961
.17 365
54
7
.82 649
.87 028
.95 622
.04378
.12 972
.17 351
53
8
.82 663
.87 016
.95 647
.04353
.12984
.17 337
52
9
.82 677
.87 005
.95 672
.04 328
.12 995
.17 323
51
10
9.82 691
9.86 993
9.95 698
0.04 302
0.13 007
0.17 309
50
11
.82 705
.86 982
.95 723
.04 277
.13018
.17 295
49
12
.82 719
.86 970
.95 748
.04252
.13 030
.17 281
48
13
.82 733
.86 959
.95 774
.04 226
.13 041
.17 267
47
14
.82 747
.86 947
.95 799
.04201
.13 053
.17 253
46
15
9.82 761
9.86 936
9.95 825
0.04 175
0.13 064
0.17 239
45
16
.82 775
.86 924
.95 850
.04150
.13 076
.17 225
44
17
.82 788
.86 913
.95 875
.04125
.13 087
.17212
43
18
.82 802
.86 902
.95 901
.04 099
.13 098
.17 198
42
19
.82 816
.86 890
.95 926
.04074
.13 110
.17 184
41
20
9.82 830
9.86 879
9.95 952
0.04 048
0.13 121
0.17 170
40
21
.82 844
.86 867
.95 977
.04 023
.13 133
.17 156
39
22
.82 858
.86 855
.96 002
.03 998
.13 145
.17 142
38
23
.82 872
.86844
.96 028
.03972
.13 156
.17 128
37
24
.82 885
.86 832
.96 053
.03 947
.13 168
.17 115
36
25
9.82 899
9.86 821
9.96 078
0.03 922
0.13 179
0.17 101
35
26
.82 913
.86 809
.96 104
.03 896
.13 191
.17 087
34
27
.82 927
.86 798
.96 129
.03 871
.13 202
.17 073
33
28
.82 941
.86 786
.96 155
.03 845
.13214
.17 059
32
29
.82 955
.86 775
.96 180
.03 820
.13 225
.17 045
31
30
9.82 968
9.86 763
9.96 205
0.03 795
0.13 237
0.17 032
30
31
.82 982
.86 752
.96 231
.03 769
.13 248
.17018
29
32
.82 996
.86 740
.96 256
.03 744
.13 260
.17 004
28
33
.83 010
.86 728
.96 281
.03 719
.13 272
.16 990
27
34
.83023
.86 717
.96 307
.03 693
.13283
.16977
26
35
9.83 037
9.86 705
9.96 332
0.03 668
0.13 295
0.16 963
25
36
.83 051
.86 694
.96 357
.03 643
.13 306
.16 949
24
37
.83065
.86 682
.96383
.03 617
.13318
.16 935
23
38
.83 078
.86 670
.96 408
.03 592
.13 330
.16 922
22
39
.83092
.86 659
.96 433
.03 567
.13 341
.16 908
21
40
9.83 106
9.86 647
9.96 459
0.03 541
0.13 353
0.16 894
20
41
.83 120
.86 635
.96 484
.03 516
.13 365
.16 880
19
42
.83 133
.86 624
.96 510
.03 490
.13 376
.16 867
18
43
.83 147
.86 612
.96 535
.03 465
.13 388
.16 853
17
44
.83 161
.86 600
.96 560
.03 440
.13 400
.16 839
16
45
9.83 174
9.86 589
9.96 586
0.03 414
0.13411
0.16 826
15
46
.83 188
.86 577
.96611
.03 389
.13 423
.16812
14
47
.83 202
.86 565
.96 636
.03 364
.13 435
.16 798
13
48
.83 215
.86 554
.96 662
.03 338
.13 446
.16 785
12
49
.83229
.86 542
.96 687
.03 313
.13 458
.16771
11
50
9.83 242
9.86 530
9.96 712
0.03 288
0.13 470
0.16 758
10
51
.83 256
.86 518
.96 738
.03 262
.13 482
.16 744
9
52
.83 270
.86 507
.96 763
.03 237
.13 493
.16 730
8
53
.83283
.86 495
.96 788
.03 212
.13 505
.16717
7
54
.83 297
.86 483
.96 814
.03 186
.13517
.16 703
6
55
9.83 310
9.86 472
9.96 839
0.03 161
0.13 528
0.16 690
5
56
.83 324
.86 460
.96864
.03 136
.13 540
.16 676
4
57
.83 338
.86 448
.96 890
.03 110
.13 552
.16 662
3
58
.83 351
.86 436
.96 915
.03 085
.13564
.16649
2
59
.83 365
.86 425
.96 940
.03 060
.13575
.16 635
1
60
9.83 378
9.86 413
9.96 966
0.03 034
0.13 587
0.16 622
0
Cos
Sin
Cot
Tan
Csc
Sec
'
132° (312°)
(227°) 47°
Table 4. Trigonometric Logarithms
239
43° (223°)
(316°) 136°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.83 378
9.86413
9.96 966
0.03 034
0.13587
0.16622
60
1
.83 392
.86 401
.96 991
.03 009
.13 599
.16 608
59
2
.83405
.86 389
.97 016
.02984
.13611
.16 595
58
3
.83 419
.86 377
.97 042
.02 958
.13 623
.16 581
57
4
.83 432
.86366
.97 067
.02 933
.13 634
.16 568
56
5
9.83 446
9.86 354
9.97 092
0.02 908
0.13 646
0.16 554
55
6
.83459
.86 342
.97 118
.02 882
.13 658
.16 541
54
7
.83473
.86 330
.97 143
.02 857
.13 670
.16 527
53
8
.83486
.86 318
.97 168
.02 832
.13682
.16 514
52
9
.83500
.86 306
.97 193
.02 807
.13 694
.16500
51
10
9.83 513
9.86 295
9.97 219
0.02 781
0.13 705
0.16487
50
11
.83527
.86283
.97 244
.02 756
.13717
.16473
49
12
.83 540
.86271
.97 269
.02 731
.13 729
.16 460
48
13
.83554
.86 259
.97 295
.02 705
.13 741
.16 446
47
14
.83567
.86 247
.97 320
.02 680
.13 753
.16433
46
15
9.83 581
9.86 235
9.97 345
0.02 655
0.13 765
0.16419
45
16
.83594
.86 223
.97 371
.02 629
.13 777
.16 406
44
17
.83608
.86211
.97 396
.02604
.13 789
.16 392
43
18
.83621
.86 200
.97 421
.02 579
.13 800
.16 379
42
19
.83634
.86 188
.97 447
.02 553
.13812
.16 366
41
20
9.83 648
9.86 176
9.97 472
0.02 528
0.13 824
0.16352
40
21
.83661
.86164
.97 497
.02 503
.13 836
.16 339
39
22
.83674
.86 152
.97 523
.02 477
.13848
.16326
38
23
.83688
.86 140
.97 548
.02 452
.13 860
.16312
37
24
.83701
.86 128
.97 573
.02 427
.13 872
.16 299
36
25
9.83 715
9.86 116
9.97 598
0.02 402
0.13 884
0.16285
35
26
.83728
.86104
.97 624
.02 376
.13 896
.16 272
34
27
.83741
.86 092
.97 649
.02 351
.13 908
.16 259
33
28
.83755
.86 080
.97 674
.02 326
.13 920
.16 245
32
29
.83768
.86 068
.97 700
.02 300
.13 932
.16232
31
30
9.83 781
9.86 056
9.97 725
0.02 275
0.13 944
0.16219
30
31
.83 795
.86044
.97 750
.02 250
.13 956
.16 205
29
32
.83 808
.86 032
.97 776
.02 224
.13 968
.16 192
28
33
.83 821
.86 020
.97 801
.02 199
.13 980
.16 179
27
34
.83834
.86 008
.97 826
.02 174
.13 992
.16 166
26
35
9.83848
9.85 996
9.97 851
0.02 149
0.14 004
0.16 152
25
36
.83861
.85984
.97 877
.02 123
.14016
.16 139
24
37
.83874
.85972
.97 902
.02 098
.14 028
.16 126
23
38
.83887
.85 960
.97 927
.02 073
.14 040
.16 113
22
39
.83901
.85948
.97 953
.02 047
.14 052
.16 099
21
40
9.83 914
9.85 936
9.97 978
0.02 022
0.14 064
0.16 086
20
41
.83927
.85924
.98 003
.01 997
.14 076
.16 073
19
42
.83940
.85912
.98 029
.01 971
.14 088
.16 060
18
43
.83954
.85900
.98054
.01 946
.14 100
.16 046
17
44
.83967
.85888
.98 079
.01 921
.14 112
.16 033
16
45
9.83 980
9.85 876
9.98 104
0.01 896
0.14 124
0.16020
15
46
.83993
.85864
.98 130
.01 870
.14 136
.16 007
14
47
.84006
.85851
.98 155
.01845
.14 149
.15 994
13
48
.84020
.85839
.98 180
.01 820
.14 161
.15 980
12
49
.84033
.85827
.98 206
.01 794
.14 173
.15 967
11
50
9.84046
9.85 815
9.98 231
0.01 769
0.14 185
0.15 954
10
51
.84059
.85 803
.98 256
.01 744
.14 197
.15 941
9
52
.84072
.85791
.98 281
.01 719
.14 209
.15 928
8
53
.84085
.85 779
.98 307
.01 693
.14 221
.15915
7
54
.84098
.85766
.98 332
.01 668
.14 234
.15 902
6
55
9.84 112
9.85 754
9.98 357
0.01 643
0.14 246
0.15 888
5
56
.84 125
.85742
.98 383
.01 617
.14 258
.15 875
4
57
.84138
.85730
.98 408
.01 592
.14 270
.15862
3
58
.84151
.85718
.98 433
.01 567
.14 282
.15849
2
59
.84 164
.85706
.98 458
.01 542
.14 294
.15836
1
60
9.84 177
9.85 693
9.98 484
0.01 516
0.14 307
0.15 823
0
Cos
Sin
Cot
Tan
Csc
S<-«-
'
133° (313°)
(226°) 46°
240
Table 4. Trigonometric Logarithms
44° (224°)
(315°) 135°
'
Sin
Cos
Tan
Cot
Sec
Csc
0
9.84 177
9.85 693
9.98 484
0.01 516
0.14 307
0.15 823
60
1
.84 190
.85 681
.98 509
.01 491
.14319
.15810
59
2
.84203
.85 669
.98 534
.01 466
.14 331
.15 797
58
3
.84216
.85 657
.98 560
.01 440
.14 343
.15784
57
4
.84229
.85645
.98 585
.01 415
.14 355
.15 771
56
5
9.84 242
9.85 632
9.98 610
0.01 390
0.14 368
0.15 758
55
6
.84 255
.85 620
.98 635
.01 365
.14 380
.15 745
54
7
.84269
.85 608
.98 661
.01 339
.14 392
.15 731
53
8
.84282
.85 596
.98 686
.01 314
.14 404
.15718
52
9
.84295
.85 583
.98 711
.01 289
.14417
.15 705
51
10
9.84 308
9.85 571
9.98 737
0.01 263
0.14 429
0.15 692
50
11
.84 321
.85 559
.98 762
.01 238
.14 441
.15 679
49
12
.84 334
.85 547
.98 787
.01 213
.14 453
.15 666
48
13
.84347
.85 534
.98 812
.01 188
.14 466
.15 653
47
14
.84360
.85 522
.98 838
.01 162
.14478
.15 640
46
15
9.84 373
9.85 510
9.98 863
0.01 137
0.14 490
0.15 627
45
16
.84 385
.85 497
.98 888
.01 112
.14 503
.15615
44
17
.84 398
.85485
.98 913
.01 087
.14515
.15 602
43
18
.84411
.85 473
.98 939
.01 061
.14 527
.15 589
42
19
.84424
.85 460
.98 964
.01 036
.14 540
.15 576
41
20
9.84 437
9.85 448
9.98 989
0.01 Oil
0.14 552
0.15 563
40
21
.84450
.85 436
.99 015
.00 985
.14 564
.15 550
39
22
.84463
.85 423
.99 040
.00 960
.14 577
.15537
38
23
.84476
.85411
.99 065
.00 935
.14 589
.15 524
37
24
.84 489
.85 399
.99 090
.00 910
.14 601
.15511
36
25
9.84 502
9.85 386
9.99 116
0.00 884
0.14 614
0.15 498
35
26
.84 515
.85 374
.99 141
.00859
.14 626
.15485
34
27
.84528
.85 361
.99 166
.00 834
.14 639
.15 472
33
28
.84540
.85 349
.99 191
.00 809
.14651
.15 460
32
29
.84553
.85 337
.99 217
.00 783
.14 663
.15 447
31
30
9.84 566
9.85 324
9.99 242
0.00 758
0.14 676
0.15 434
30
31
.84579
.85 312
.99 267
.00 733
.14 688
.15421
29
32
.84592
.85 299
.99 293
.00 707
.14 701
.15 408
28
33
.84 605
.85 287
.99 318
.00 682
.14713
.15 395
27
34
.84618
.85274
.99 343
.00 657
.14 726
,15 382
26
35
9.84 630
9.85 262
9.99 368
0.00 632
0.14 738
0.15 370
25
36
.84643
.85 250
.99 394
.00 606
.14 750
.15 357
24
37
.84656
.85237
.99 419
.00 581
.14 763
.15 344
23
38
.84669
.85 225
.99 444
.00 556
.14 775
.15331
22
39
.84682
.85212
.99 469
.00 531
.14788
.15318
21
40
9.84 694
9.85 200
9.99 495
0.00 505
0.14 800
0.15 306
20
41
.84707
.85 187
.99 520
.00 480
.14813
.15 293
19
42
.84720
.85 175
.99 545
.00 455
.14 825
.15 280
18
43
.84733
.85 162
.99 570
.00 430
.14 838
.15 267
17
44
.84745
.85 150
.99 596
.00 404
.14 850
.15 255
16
45
9.84 758
9.85 137
9.99 621
0.00 379
0.14 863
0.15242
15
46
.84771
.85 125
.99 646
.00 354
.14 875
.15 229
14
47
.84784
.85 112
.99 672
.00 328
.14 888
.16216
13
48
.84796
.85 100
.99 697
.00303
.14 900
.15 204
12
49
.84809
.85 087
.99 722
.00 278
.14 913
.15 191
11
50
9.84 822
9.85 074
9.99 747
0.00 253
0.14 926
0.15 178
10
51
.84835
.85062
.99 773
.00 227
.14 938
.15 165
9
52
.84847
.85 049
.99 798
.00 202
.14 951
.15 153
8
53
.84860
.85 037
.99 823
.00 177
.14 963
.15 140
7
54
.84873
.85024
.99848
.00 152
.14 976
.15 127
6
55
9.84 885
9.85012
9.99 874
0.00 126
0.14 988
0.15 115
5
56
.84898
.84999
.99 899
.00 101
.15 001
.15 102
4
57
.84911
.84 986
.99 924
.00 076
.15 014
.15 089
3
58
.84923
.84 974
.99 949
.00 051
.15 026
.15 077
2
59
.84936
.84961
.99 975
.00 025
.15 039
.15 064
1
60
9.84 949
9.84 949
0.00 000
0.00 000
0.15 051
0.15051
0
Cos
Sin
Cot
Tan
Csc
Sec
'
134° (314°)
(225°) 45°
Table 5. Meridional Parts
241
'
0°
1°
2°
3°
4°
5°
6°
7°
8°
9°
'
0
0.0
59.6
119.2
178.9
238.6
298.3
358.2
418.2
478.3
538.6
0
1
1.0
60.6
20.2
79.9
39.6
99.3
59.2
19.2
79.3
39.6
1
2
2.0
61.6
21.2
80.8
40.6
300.3
60.2
20.2
80.3
40.6
2
3
3.0
62.6
22.2
81.8
41.6
01.3
61.2
21.2
81.3
41.6
3
4
4.0
63.6
23.2
82.8
42.5
02.3
62.2
22.2
82.3
42.6
4
5
5.0
64.6
124.2
183.8
243.5
303.3
363.2
423.2
483.3
543.6
5
6
6.0
65.6
25.2
84.8
44.5
04.3
64.2
24.2
84.3
44.6
6
7
7.0
66.5
26.2
85.8
45.5
05.3
65.2
25.2
85.3
45.6
7
8
7.9
67.5
27.2
86.8
46.5
06.3
66.2
26.2
86.3
46.6
' 8
9
8.9
68.5
28.2
87.8
47.5
07.3
67.2
27.2
87.3
47.6
9
10
9.9
69.5
129.1
188.8
248.5
308.3
368.2
428.2
488.3
548.6
10
11
10.9
70.5
30.1
89.8
49.5
09.3
69.2
29.2
89.3
49.6
11
12
11.9
71.5
31.1
90.8
50.5
10.3
70.2
30.2
90.4
50.6
12
13
12.9
72.5
32.1
91.8
51.5
11.3
71.2
31.2
91.4
51.7
13
14
13.9
73.5
33.1
92.8
52.5
12.3
72.2
32.2
92.4
52.7
14
15
14.9
74.5
134.1
193.8
253.5
313.3
373.2
433.2
493.4
553.7
15
16
15.9
75.5
35.1
94.8
54.5
14.3
74.2
34.2
94.4
54.7
16
17
16.9
76.5
36.1
95.8
55.5
15.3
75.2
35.2
95.4
55.7
17
18
17.9
77.5
37.1
96.8
56.5
16.3
76.2
36.2
96.4
56.7
18
19
18.9
78.5
38.1
97.8
57.5
17.3
77.2
37.2
97.4
57.7
19
20
19.9
79.5
139.1
198.8
258.5
318.3
378.2
438.2
498.4
558.7
20
21
20.9
80.5
40.1
99.7
59.5
19.3
79.2
39.2
99.4
59.7
21
22
21.9
81.5
41.1
200.7
60.5
20.3
80.2
40.2
500.4
60.7
22
23
22.8
82.4
42.1
01.7
61.5
21.3
81.2
41.2
01.4
61.7
23
24
23.8
83.4
43.1
02.7
62.5
22.3
82.2
42.2
02.4
62.7
24
25
24.8
84.4
144.1
203.7
263.5
323.3
383.2
443.2
503.4
563.7
25
26
25.8
85.4
45.1
04.7
64.5
24.3
84.2
44.2
04.4
64.7
26
27
26.8
86.4
46.0
05.7
65.5
25.3
85.2
45.2
05.4
65.7
27
28
27.8
87.4
47.0
06.7
66.5
26.3
86.2
46.2
06.4
66.8
28
29
28.8
88.4
48.0
07.7
67.4
27.3
87.2
47.2
07.4
67.8
29
30
29.8
89.4
149.0
208.7
268.4
328.3
388.2
448.2
508.4
568.8
30
31
30.8
90.4
50.0
09.7
69.4
29.3
89.2
49.2
09.4
69.8
31
32
31.8
91.4
51.0
10.7
70.4
30.3
90.2
50.2
10.4
70.8
32
33
32.8
92.4
52.0
11.7
71.4
31.3
91.2
51.2
11.4
71.8
33
34
33.8
93.4
53.0
12.7
72.4
32.3
92.2
52.2
12.4
72.8
34
35
34.8
94.4
154.0
213.7
273.4
333.3
393.2
453.2
513.4
573.8
35
36
35.8
95.4
55.0
14.7
74.4
34.3
94.2
54.3
14.5
74.8
36
37
36.7
96.4
56.0
15.7
75.4
35.3
95.2
55.3
15.5
75.8
37
38
37.7
97.3
57.0
16.7
76.4
36.2
96.2
56.3
16.5
76.8
38
39
38.7
98.3
58.0
17.7
77.4
37.2
97.2
57.3
17.5
77.8
39
40
39.7
99.3
159.0
218.7
278.4
338.2
398.2
458.3
518.5
578.8
40
41
40.7
100.3
60.0
19.7
79.4
39.2
99.2
59.3
19.5
79.9
41
42
41.7
01.3
61.0
20.6
80.4
40.2
400.2
60.3
20.5
80.9
42
43
42.7
02.3
62.0
21.6
81.4
41.2
01.2
61.3
21.5
81.9
43
44
43.7
03.3
63.0
22.6
82.4
42.2
02.2
62.3
22.5
82.9
44
45
44.7
104.3
164.0
223.6
283.4
343.2
403.2
463.3
523.5
583.9
45
46
45.7
05.3
65.0
24.6
84.4
44.2
04.2
64.3
24.5
84.9
46
47
46.7
06.3
66.0
25.6
85.4
45.2
05.2
65.3
25.5
85.9
47
48
47.7
07.3
67.0
26.6
86.4
46.2
06.2
66.3
26.5
86.9
48
49
48.7
08.3
68.0
27.6
87.4
47.2
07.2
67.3
27.5
87.9
49
50
49.7
109.3
168.9
228.6
288.4
348.2
408.2
468.3
528.5
588.9
50
51
50.7
10.3
69.9
29.6
89.4
49.2
09.2
69.3
29.5
89.9
51
52
51.6
11.3
70.9
30.6
90.4
50.2
10.2
70.3
30.5
90.9
52
53
52.6
12.3
71.9
31.6
91.4
51.2
11.2
71.3
31.5
91.9
53
54
53.6
13.2
72.9
32.6
92.4
52.2
12.2
72.3
32.5
93.0
54
55
54.6
114.2
173.9
233.6
293.4
353.2
413.2
473.3
533.5
594.0
55
56
55.6
15.2
74.9
34.6
94.4
54.2
14.2
74.3
34.6
95.0
56
57
56.6
16.2
75.9
35.6
95.4
55.2
15.2
75.3
35.6
96.0
57
58
57.6
17.2
76.9
36.6
96.3
56.2
16.2
76.3
36.6
97.0
58
59
58.6
18.2
77.9
37.6
97.3
57.2
17.2
77.3
37.6
98.0
59
60
59.6
119.2
178.9
238.6
298.3
358.2
418.2
478.3
538.6
599.0
60
'
0°
1°
2°
3°
4°
5°
6°
7°
8°
9°
/
242
Table 5. Meridional Parts
/
10°
11°
12°
13°
14°
15°
16°
17°
18°
19°
'
0
599.0
659.6
720.5
781.5
842.8
904.4
966.3
1028.5
1091.0
1153.9
0
1
600.0
60.6
21.5
82.5
43.9
05.4
67.3
29.5
92.0
54.9
1
2
01.0
61.7
22.5
83.6
44.9
06.5
68.3
30.5
93.1
56.0
2
3
02.0
62.7
23.5
84.6
45.9
07.5
69.4
31.6
94.1
57.0
3
4
03.0
63.7
24.5
85.6
46.9
08.5
70.4
32.6
95.2
58.1
4
5
604.1
664.7
725.5
786.6
847.9
909.6
971.4
1033.7
1096.2
1159.1
5
6
05.1
65.7
26.6
87.6
49.0
10.6
72.5
34.7
97.3
60.2
6
7
06.1
66.7
27.6
88.7
50.0
11.6
73.5
35.7
98.3
61.2
7
8
07.1
67.7
28.6
89.7
51.0
12.6
74.6
36.8
99.4
62.3
8
9
08.1
68.7
29.6
90.7
52.0
13.7
75.6
37.8
1100.4
63.3
9
10
609.1
669.8
730.6
791.7
853.1
914.7
976.6
1038.9
1101.4
1164.4
10
11
10.1
70.8
31.6
92.7
54.1
15.7
77.7
39.9
02.5
65.4
11
12
11.1
71.8
32.7
93.8
55.1
16.8
78.7
40.9
03.5
66.5
12
13
12.1
72.8
33.7
94.8
56.1
17.8
79.7
42.0
04.6
67.5
13
14
13.1
73.8
34.7
95.8
57.2
18.8
80.8
43.0
05.6
68.6
14
15
614.1
674.8
735.7
796.8
858.2
919.8
981.8
1044.1
1106.7
1169.7
15
16
15.2
75.8
36.7
97.8
59.2
20.9
82.8
45.1
07.7
70.7
16
17
16.2
76.8
37.7
98.9
60.2
21.9
83.9
46.1
08.8
71.8
17
18
17.2
77.9
38.8
99.9
61.3
22.9
84.9
47.2
09.8
72.8
18
19
18.2
78.9
39.8
800.9
62.3
24.0
85.9
48.2
10.9
73.9
19
20
619.2
679.9
740.8
801.9
863.3
925.0
987.0
1049.3
1111.9
1174.9
20
21
20.2
80.9
41.8
02.9
64.3
26.0
88.0
50.3
13.0
76.0
21
22
21.2
81.9
42.8
04.0
65.4
27.1
89.0
51.3
14.0
77.0
22 -
23
22.2
82.9
43.8
05.0
66.4
28.1
90.1
52.4
15.0
78.1
23
24
23.2
83.9
44.9
06.0
67.4
29.1
91.1
53.4
16.1
79.1
24
25
624.2
684.9
745.9
807.0
868.5
930.1
992.1
1054.5
1117.1
1180.2
25
26
25.3
86.0
46.9
08.1
69.5
31.2
93.2
55.5
18.2
81.2
26
27
26.3
87.0
47.9
09.1
70.5
32.2
94.2
56.6
19.2
82.3
27
28
27.3
88.0
48.9
10.1
71.5
33.2
95.3
57.6
20.3
83.3
28
29
28.3
89.0
49.9
11.1
72.6
34.3-
96.3
58.6
21.3
84.4
29
30
629.3
690.0
751.0
812.1
873.6
935.3
997.3
1059.7
1122.4
1185.5
30
31
30.3
91.0
52.0
13.2
74.6
36.3
98.4
60.7
23.4
86.5
31
32
31.3
92.0
53.0
14.2
75.6
37.4
99.4
61.8
24.5
87.6
32
33
32.3
93.1
54.0
15.2
76.7
38.4
1000.4
62.8
25.5
88.6
33
34
33.3
94.1
55.0
16.2
77.7
39.4
01.5
63.9
26.6
89.7
34
35
634.3
695.1
756.0
817.3
878.7
940.5
1002.5
1064.9
1127.6
1190.7
35
36
35.4
96.1
57.1
18.3
79.7
41.5
03.6
65.9
28.7
91.8
36
37
36.4
97.1
58.1
19.3
80.8
42.5
04.6
67.0
29.7
92.8
37
38
37.4
98.1
59.1
20.3
81.8
43.6
05.6
68.0
30.8
93.9
38
39
38.4
99.1
60.1
21.3
82.8
44.6
06.7
69.1
31.8
95.0
39
40
639.4
700.2
761.1
822.4
883.8
945.6
1007.7
1070.1
1132.9
1196.0
40
41
40.4
01.2
62.2
23.4
84.9
46.7
08.7
71.2
33.9
97.1
41
42
41.4
02.2
63.2
24.4
85.9
47.7
09.8
72.2
35.0
98.1
42
43
42.4
03.2
64.2
25.4
86.9
48.7
10.8
73.2
36.0
99.2
43
44
43.4
04.2
65.2
26.5
88.0
49.7
11.8
74.3
37.1
1200.2
44
45
644.5
705.2
766.2
827.5
889.0
950.8
1012.9
1075.3
1138.1
1201.3
45
46
45.5
06.2
67.3
28.5
90.0
51.8
13.9
76.4
39.2
02.3
46
47
46.5-
07.3
68.3
29.5
91.0
52.8
15.0
77.4
40.2
03.4
47
48
47.5
08.3
69.3
30.5
92.1
53.9
16.0
78.5
41.8
04.5
48
49
48.5
09.3
70.3
31.6
93.1
54.9
17.0
79.5
42.3
05.5
49
50
649.5
710.3
771.3
832.6
894.1
955.9
1018.1
1080.5
1143.4
1206.6
50
51
50.5
11.3
72.3
33.6
95.2
57.0
19.1
81.6
44.4
07.6
51
52
51.5
12.3
73.4
34.6
96.2
58.0
20.2
82.6
45.5
08.7
52
53
52.5
13.4
74.4
35.7
97.2
59.0
21.2
83.7
46.5
09.7
53
54
53.6
14.4
75.4
36.7
98.2
60.1
22.2
84.7
47.6
10.8
54
55
654.6
715.4
776.4
837.7
899.3
961.1
1023.3
1085.8
1148.6
1211.8
55
56
55.6
16.4
77.4
38.7
900.3
62.1
24.3
86.8
49.7
12.9
56
57
56.6
17.4
78.5
39.8
01.3
63.2
25.3
87.9
50.7
14.0
57
58
57.6
18.4
79.5
40.8
02.3
64.2
26.4
88.9
51.8
15.0
58 '
59
58.6
19.4
80.5
41.8
03.4
65.2
27.4
89.9
52.8
16.1
59
60
659.6
720.5
781.5
842.8
904.4
966.3
1028.5
1091.0
1153.9
1217.1
60
'
10°
11°
12°
13°
14°
15°
16°
17°
18°
19°
'
Table 5. Meridional Parts
243
'
20°
21°
22°
23°
24°
25°
26°
27°
28°
29°
'
0
1217.1
1280.8
1344.9
1409.5
1474.5
1540.1
1606.2
1672.9
1740.2
1808.1
0
1
18.2
81.9
46.0
10.6
75.6
41.2
07.3
74.0
41.3
09.2
1
2
19.3
82.9
47.1
11.6
76.7
42.3
08.4
75.1
42.4
10.4
2
3
20.3
84.0
48.1
12.7
77.8
43.4
09.5
76.2
43.6
11.5
3
4
21.4
85.1
49.2
13.8
78.9
44.5
10.6
77.4
44.7
12.6
4
5
1222.4
1286.1
1350.3
1414.9
1480.0
1545.6
1611.7
1678.5
1745.8
1813.8
5
6
23.5
87.2
51.4
16.0
81.1
46.7
12.9
79.6
46.9
14.9
6
7
24.5
88.3
52.4
17.1
82.2
47.8
14.0
80.7
48.1
16.1
7
8
25.6
89.3
53.5
18.1
83.3
48.9
15.1
81.8
49.2
17.2
8
9
26.7
90.4
54.6
19.2
84.3
50.0
16.2
82.9
50.3
18.3
9
10
1227.7
1291.5
1355.7
1420.3
1485.4
1551.1
1617.3
1684.1
1751.5
1819.5
10
11
28.8
92.5
56.7
21.4
86.5
52.2
18.4
85.2
52.6
20.6
11
12
29.8
93.6
57.8
22.5
87.6
53.3
19.5
86.3
53.7
21.8
12
13
30.9
94.7
58.9
23.5
88.7
54.4
20.6
87.4
54.8
22.9
13
14
32.0
95.7
59.9
24.6
89.8
55.5
21.7
88.5
56.0
24.0
14
15
1233.0
1296.8
1361.0
1425.7
1490.9
1556.6
1622.8
1689.7
1757.1
1825.2
15
16
34.1
97.9
62.1
26.8
92.0
57.7
23.9
90.8
58.2
26.3
16
17
35.1
98.9
63.2
27.9
93.1
58.8
25.0
91.9
59.4
27.5
17
18
36.2
1300.0
64.2
29.0
94.2
59.9
26.2
93.0
60.5
28.6
18
19
37.3
01.1
65.3
30.0
95.2
61.0
27.3
94.1
61.6
29.7
19
20
1238.3
1302.1
1366.4
1431.1
1496.3
1562.1
1628.4
1695.3
1762.7
1830.9
20
21
39.4
03.2
67.5
32.2
97.4
63.2
29.5
96.4
63.9
32.0
21
22
40.4
04.3
68.5
33.3
98.5
64.3
30.6
97.5
65.0
33.2
22
23
41.5
05.3
69.6
34.4
99.6
65.4
31.7
98.6
66.1
34.3
23
24
42.6
06.4
70.7
35.4
1500.7
66.5
32.8
99.7
67.3
35.4
24
25
1243.6
1307.5
1371.8
1436.5
1501.8
1567.6
1633.9
1700.9
1768.4
1836.6
25
26
44.7
08.5
72.8
37.6
02.9
68.7
35.0
02.0
69.5
37.7
26
27
45.7
09.6
73.9
38.7
04.0
69.8
36.1
03.1
70.7
38.9
27
28
46.8
10.7
75.0
39.8
05.1
70.9
37.3
04.2
71.8
40.0
28
29
47.9
11.7
76.1
40.9
06.2
72.0
38.4
05.3
72.9
41.2
29
30
1248.9
1312.8
1377.1
1442.0
1507.3
1573.1
1639.5
1706.5
1774.1
1842.3
30
31
50.0
13.9
78.2
43.0
08.4
74.2
40.6
07.6
75.2
43.4
31
32
51.0
14.9
79.3
44.1
09.4
75.3
41.7
08.7
76.3
44.6
32
33
52.1
16.0
80.4
45.2
10.5
76.4
42.8
09.8
77.4
45.7
33
34
53.2
17.1
81.5
46.3
11.6
77.5
43.9
10.9
78.6
46.9
34
35
1254.2
1318.2
1382.5
1447.4
1512.7
1578.6
1645.0
1712.1
1779.7
1848.0
35
36
55.3
19.2
83.6
48.5
13.8
79.7
46.2
13.2
80.8
49.2
36
37
56.4
20.3
84.7
49.5
14.9
80.8
47.3
14.3
82.0
50.3
37
38
57.4
21.4
85.8
50.6
16.0
81.9
48.4
15.4
83.1
51.4
38
39
58.5
22.4
86.8
51.7
17.1
83.0
49.5
16.6
84.2
52.6
39
40
1259.5
1323.5
1387.9
1452.8
1518.2
1584.1
1650.6
1717.7
1785.4
1853.7
40
41
60.6
24.6
89.0
53.9
19.3
85.2
51.7
18.8
86.5
54.9
41
42
61.7
25.6
90.1
55.0
20.4
86.3
52.8
19.9
87.6
56.0
42
43
62.7
26.7
91.1
56.1
21.5
87.4
53.9
21.1
88.8
57.2
43
44
63.8
27.8
92.2
57.1
22.6
88.5
55.1
22.2
89.9
58.3
44
45
1264.9
1328.9
1393.3
1458.2
1523.7
1589.6
1656.2
1723.3
1791.1
1859.5
45
46
65.9
29.9
94.4
59.3
24.8
90.7
57.3
24.4
92.2
60.6
46
47
67.0
31.0
95.5
60.4
25.9
91.8
58.4
25.5
93.3
61.8
47
48
68.0
32.1
96.5
61.5
27.0
92.9
59.5
26.7
94.5
62.9
48
49
69.1
33.1
97.6
62.6
28.0
94.1
60.6
27.8
95.6
64.0
49
50
1270.2
1334.2
1398.7
1463.7
1529.1
1595.2
1661.7
1728.9
1796.7
1865.2
50
51
71.2
35.3
99.8
64.8
30.2
96.3
62.9
30.0
97.9
66.3
51
52
72.3
36.3
1400.9
65.8
31.3
97.4
64.0
31.2
99.0
67.5
52
53
73.4
37.4
01.9
66.9
32.4
98.5
65.1
32.3
1800.1
68.6
53
54
74.4
38.5
03.0
68.0
33.5
99.6
66.2
33.4
01.3
69.8
54
55
1275.5
1339.6
1404.1
1469.1
1534.6
1600.7
1667.3
1734.5
1802.4
1870.9
55
56
76.6
40.6
05.2
70.2
35.7
01.8
68.4
35.7
03.5
72.1
56
57
• 77.6
41.7
06.2
71.3
36.8
02.9
69.5
36.8
04.7
73.2
57
58
78.7
42.8
07.3
72.4
37.9
04.0
70.7
37.9
05.8
74.4
58
59
79.7
43.8
08.4
73.5
39.0
05.1
71.8
39.1
07.0
75.5
59
60
1280.8
1344.9
1409.5
1474.5
1540.1
1606.2
1672.9
1740.2
1808.1
1876.7
60
'
20°
21°
22°
23°
24°
25°
26°
27°
28°
29°
'
244
Table 5. Meridional Parts
'
30°
31°
32°
33°
34°
35°
36°
37°
38°
39°
'
0
1876.7
1946.0
2016.0
2086.8
2158.4
2230.9
2304.2
2378.5
2453.8
2530.2
0
1
77.8
47.1
17.2
88.0
59.6
32.1
05.5
79.8
55.1
31.5
1
2
79.0
48.3
18.3
89.2
60.8
33.3
06.7
81.0
56.4
32.8
2
3
80.1
49.4
19.5
90.3
62.0
34.5
07.9
82.3
57.6
34.0
3
4
81.3
50.6
20.7
91.5
63.2
35.7
09.2
83.5
58.9
35.3
4
5
1882.4
1951.8
2021.9
2092.7
2164.4
2236.9
2310.4
2384.8
2460.2
2536.6
5
6
83.6
52.9
23.0
93.9
65.6
38.2
11.6
86.0
61.4
37.9
6
7
84.7
54.1
24.2
95.1
66.8
39.4
12.9
87.3
62.7
39.2
7
8
85.9
55.3
25.4
96.3
68.0
40.6
14.1
88.5
64.0
40.5
8
9
87.0
56.4
26.6
97.5
69.2
41.8
15.3
89.8
65.2
41.7
9
10
1888.2
1957.6
2027.7
2098.7
2170.4
2243.0
2316.5
2391.0
2466.5
2543.0
10
11
89.3
58.7
28.9
99.8
71.6
44.2
17.8
92.3
67.8
44.3
11
12
90.5
59.9
30.1
2101.0
72.8
45.5
19.0
93.5
69.0
45.6
12
13
91.6
61.1
31.3
02.2
74.0
46.7
20.3
94.8
70.3
46.9
13
14
92.8
62.2
32.4
03.4
75.2
47.9
21.5
96.0
71.6
48.2
14
15
1893.9
1963.4
2033.6
2104.6
2176.4
2249.1
2322.7
2397.3
2472.8
2549.5
15
16
95.1
64.6
34.8
05.8
77.6
50.3
24.0
98.5
74.1
50.7
16
17
96.2
65.7
36.0
07.0
78.8
51.6
25.2
99.8
75.4
52.0
17
18
97.4
66.9
37.1
08.2
80.0
52.8
26.4
2401.0
76.6
53.3
18 •
19
98.5
68.1
38.3
09.4
81.2
54.0
27.7
02.3
77.9
54.6
19
20
1899.7
1969.2
2039.5
2110.6
2182.5
2255.2
2328.9
2403.5
2479.2
2555.9
20
21
1900.8
70.4
40.7
11.8
83.7
56.4
30.1
04.8
80.4
57.2
21
22
02.0
71.5
41.8
12.9
84.9
57.7
31.4
06.0
81.7
58.5
22
23
03.1
72.7
43.0
14.1
86.1
58.9
32.6
07.3
83.0
59.8
23
24
04.3
73.9
44.2
15.3
87.3
60.1
33.8
08.5
84.3
61.0
24
25
1905.5
1975.0
2045.4
2116.5
2188.5
2261.3
2335.1
2409.8
2485.5
2562.3
25
26
06.6
76.2
46.6
17.7
89.7
62.5
36.3
11.1
86.8
63.6
26
27
07.8
77.4
47.7
18.9
90.9
63.8
37.6
12.3
88.1
64.9
27
28
08.9
78.5
48.9
20.1
92.1
65.0
38.8
13.6
89.3
66.2
28
29
10.1
79.7
50.1
21.3
93.3
66.2
40.0
14.8
90.6
67.5
29
30
1911.2
1980.9
2051.3
2122.5
2194.5
2267.4
2341.3
2416.1
2491.9
2568.8
30
31
12.4
82.0
52.5
23.7
95.7
68.7
42.5
17.3
93.2
70.1
31
32
13.5
83.2
53.6
24.9
96.9
69.9
43.7
18.6
94.4
71.4
32
33
14.7
84.4
54.8
26.1
98.1
71.1
45.0
19.8
95.7
72.7
33
34
15.8
85.5
56.0
27.3
99.4
72.3
46.2
21.1
97.0
73.9
34
35
1917.0
1986.7
2057.2
2128.5
2200.6
2273.5
2347.5
2422.3
2498.3
2575.2
35
36
18.2
87.9
58.4
29.6
01.8
74.8
48.7
23.6
99.5
76.5
36
37
19.3
89.1
59.5
30.8
03.0
76.0
49.9
24.9
2500.8
77.8
37
38
20.5
90.2
60.7
32.0
04.2
77.2
51.2
26.1
02.1
79.1
38
39
21.6
91.4
61.9
33.2
05.4
78.4
52.4
27.4
03.4
80.4
39
40
1922.8
1992.6
2063.1
2134.4
2206.6
2279.7
2353.7
2428.6
2504.6
2581.7
40
41
23.9
93.7
64.3
35.6
07.8
80.9
54.9
29.9
05.9
83.0
41
42
25.1
94.9
65.5
36.8
09.0
82.1
56.1
31.2
07.2
84.3
42
43
26.3
96.1
66.6
38.0
10.2
83.3
57.4
32.4
08.5
85.6
43
44
27.4
97.2
67.8
39.2
11.5
84.6
58.6
33.7
09.7
86.9
44
45
1928.6
1998.4
2069.0
2140.4
2212.7
2285.8
2359.9
2434.9
2511.0
2588.2
45
46
29.7
99.6
70.2
41.6
13.9
87.0
61.1
36.2
12.3
89.5
46
47
30.9
2000.7
71.4
42.8
15.1
88.3
62.4
37.4
13.6
90.8
47
48
32.0
01.9
72.6
44.0
16.3
89.5
63.6
38.7
14.8
92.1
48
49
33.2
03.1
73.7
45.2
17.5
90.7
64.8
40.0
16.1
93.4
49
50
1934.4
2004.3
2074.9
2146.4
2218.7
2291.9
2366.1
2441.2
2517.4
2594.7
50
51
35.5
05.4
76.1
47.6
19.9
93.2
67.3
42.5
18.7
96.0
51
52
36.7
06.6
77.3
48.8
21.1
94.4
68.6
43.7
20.0
97.3
52
53
37.8
07.8
78.5
50.0
22.4
95.6
69.8
45.0
21.2
98.5
53
54
39.0
08.9
79.7
51.2
23.6
96.9
71.1
46.3
22.5
99.8
54
55
1940.2
2010.1
2080.8
2152.4
2224.8
2298.1
2372.3
2447.5
2523.8
2601.1
55
56
41.3
11.3
82.0
53.6
26.0
99.3
73.6
48.8
25.1
02.4
56
57
42.5
12.5
83.2
54.8
27.2
2300.5
74.8
50.1
26.4
03.7
57
58
43.6
13.6
84.4
56.0
28.4
01.8
76.1
51.3
27.6
05.0
58
59
44.8
14.8
85.6
57.2
29.6
03.0
77.3
52.6
28.9
06.3
59
60
1946.0
2016.0
JOSO.M
2158.4
2230.9
2304.2
2378.5
2453.8
2530.2
2607.6
60
> i /
L.J
30°
31°
32°
33°
34°
35°
36°
37°
38°
39°
'
Table 5. Meridional Parts
245
'
40°
41°
42°
43°
44°
45°
46°
47°
48°
49°
'
0
2607.6
2686.2
2766.0
2847.1
2929.5
3013.4
3098.7
3185.6
3274.1
3364.4
0
1
08.9
87.6
67.4
48.5
30.9
14.8
3100.1
87.1
75.6
65.9
1
2
10.2
88.9
68.7
49.9
32.3
16.2
01.6
88.5
77.1
67.4
2
3
11.5
90.2
70.1
51.2
33.7
17.6
03.0
90.0
78.6
69.0
3
4
12.8
91.5
71.4
52.6
35.1
19.0
04.4
91.4
80.1
70.5
4
5
2614.1
2692.8
2772.8
2853.9
2936.5
3020.4
3105.9
3192.9
3281.6
3372.0
5
6
15.4
94.2
74.1
55.3
37.9
21.8
07.3
94.4
83.1
73.5
6
7
16.8
95.5
75.4
56.7
39.3
23.3
08.8
95.8
84.6
75.1
7
8
18.1
96.8
76.8
58.0
40.6
24.7
10.2
97.3
86.1
76.6
8
9
19.4
98.1
78.1
59.4
42.0
26.1
11.6
98.8
87.6
78.1
9
10
2620.7
2699.5
2779.5
2860.8
2943.4
3027.5
3113.1
3200.2
3289.0
3379.6
10
11
22.0
2700.8
80.8
62.1
44.8
28.9
14.5
01.7
90.5
81.2
11
12
23.3
02.1
82.2
63.5
46.2
30.3
16.0
03.2
92.0
82.7
12
13
24.6
03.4
83.5
64.9
47.6
31.7
17.4
04.6
93.5
84.2
13
14
25.9
04.8
84.8
66.2
49.0
33.2
18.8
06.1
95.0
85.7
14
15
2627.2
2706.1
2786.2
2867.6
2950.4
3034.6
3120.3
3207.6
3296.5
3387.3
15
16
28.5
07.4
87.5
69.0
51.8
36.0
21.7
09.0
98.0
88.8
16
17
29.8
08.7
88.9
70.3
53.2
37.4
23.2
10.5
99.5
90.3
17
18
31.1
10.1
90.2
71.7
54.5
38.8
24.6
12.0
3301.0
91.8
18
19
32.4
11.4
91.6
73.1
55.9
40.2
26.0
13.4
02.5
93.4
19
20
2633.7
2712.7
2792.9
2874.4
2957.3
3041.7
3127.5
3214.9
3304.0
3394.9
20
21
35.0
14.0
94.3
75.8
58.7
43.1
28.9
16.4
05.5
96.4
21
22
36.3
15.4
95.6
77.2
60.1
44.5
30.4
17.9
07.0
98.0
22
23
37.6
16.7
97.0
78.6
61.5
45.9
31.8
19.3
08.5
99.5
23
24
38.9
18.0
98.3
79.9
62.9
47.3
33.3
20.8
10.0
3401.0
24
25
2640.2
2719.3
2799.7
2881.3
2964.3
3048.7
3134.7
3222.3
3311.5
3402.6
25
26
41.6
20.7
2801.0
82.7
65.7
50.2
36.2
23.7
13.0
04.1
26
27
42.9
22.0
02.4
84.0
67.1
51.6
37.6
25.2
14.5
05.6
27
28
44.2
23.3
03.7
85.4
68.5
53.0
39.0
26.7
16.0
07.2
28
29
45.5
24.7
05.1
86.8
69.9
54.4
40.5
28.2
17.5
08.7
29
30
2646.8
2726.0
2806.4
2888.2
2971.3
3055.9
3141.9
3229.6
3319.0
3410.2
30
31
48.1
27.3
07.8
89.5
72.7
57.3
43.4
31.1
20.5
11.8
31
32
49.4
28.6
09.1
90.9
74.1
58.7
44.8
32.6
22.1
13.3
32
33
50.7
30.0
10.5
92.3
75.5
60.1
46.3
34.1
23.6
14.8
33
34
52.0
31.3
11.8
93.7
76.9
61.5
47.7
35.6
25.1
16.4
34
35
2653.3
2732.6
2813.2
2895.0
2978.3
3063.0
3149.2
3237.0
3326.6
3417.9
35
36
54.7
34.0
14.5
96.4
79.7
64.4
50.6
38.5
28.1
19.5
36
37
56.0
35.3
15.9
97.8
81.1
65.8
52.1
40.0
29.6
21.0
37
38
57.3
36.6
17.2
99.2
82.5
67.2
53.5
41.5
31.1
22.5
38
39
58.6
38.0
18.6
2900.5
83.9
68.7
55.0
42.9
32.6
24.1
39
40
2659.9
2739.3
2820.0
2901.9
2985.3
3070.1
3156.4
3244.4
3334.1
3425.6
40
41
61.2
40.6
21.3
03.3
86.7
71.5
57.9
45.9
35.6
27.2
41
42
62.5
42.0
22.7
04.7
88.1
72.9
59.4
47.4
37.1
28.7
42
43
63.9
43.3
24.0
06.1
89.5
74.4
60.8
48.9
38.6
30.2
43
44
65.2
44.6
25.'4
07.4
90.9
75.8
62.3
50.3
40.2
31.8
44
45
2666.5
2746.0
2826.7
2908.8
2992.3
3077.2
3163.7
3251.8
3341.7
3433.3
45
46
67.8
47.3
28.1
10.2
93.7
78.7
65.2
53.3
43.2
34.9
46
47
69.1
48.6
29.4
11.6
95.1
80.1
66.6
54.8
44.7
36.4
47
48
70.4
50.0
30.8
13.0
96.5
81.5
68.1
56.3
46.2
38.0
48
49
71.7
51.3
32.2
14.3
97.9
82.9
69.5
57.8
47.7
39.5
49
50
2673.1
2752.7
2833.5
2915.7
2999.3
3084.4
3171.0
3259.3
3349.2
3441.0
50
51
74.4
54.0
34.9
17.1
3000.7
85.8
72.5
60.7
50.8
42.6
51
52
75.7
55.3
36.2
18.5
02.1
87.2
73.9
62.2
52.3
44.1
52
53
77.0
56.7
37.6
19.9
03.5
88.7
75.4
63.7
53.8
45.7
53
54
78.3
58.0
39.0
21.2
04.9
90.1
76.8
65.2
55.3
47.2
54
55
2679.6
2759.3
2840.3
2922.6
3006.3
3091.5
3178.3
3266.7
3356.8
3448.8
55
56
81.0
60.7
41.7
24.0
07.7
93.0
79.7
68.2
58.3
5Q.3
56
57
82.3
62.0
43.0
25.4
09.2
94.4
81.2
69.7
59.9
51.9
57
58
83.6
63.4
44.4
26.8
10.6
95.8
82.7
71.1
61.4
53.4
58
59
84.9
64.7
45.8
28.2
12.0
97.3
84.1
72.6
62.9
55.0
59
60
2686.2
2766.0
2847.1
2929.5
3013.4
3098.7
3185.6
3274.1
3364.4
3456.5
60
'
40°
41°
42°
43°
44°
45°
46°
47°
48°
49°
,.<*
m
246
Table 5. Meridional Parts
'
50°
51°
52°
53°
54°
55°
56°
57°
58°
59°
/
0
3456.5
3550.6
3646.7
3745.1
3845.7
3948.8
4054.5
4163.0
4274.4
4389.1
0
1
58.1
52.2
48.4
46.7
47.4
50.5
56.3
64.8
76.3
91.0
1
2
59.6
53.8
50.0
48.4
49.1
52.3
58.1
66.6
78.2
92.9
2
3
61.2
55.4
51.6
50.0
50.8
54.0
59.8
68.5
80.1
94.9
3
4
62.7
56.9
53.2
51.7
52.5
55.7
61.6
70.3
82.0
96.8
4
5
3464.3
3558.5
3654.8
3753.4
3854.2
3957.5
4063.4
4172.1
4283.9
4398.8
5
6
65.9
60.1
56.5
55.0
55.9
59.2
65.2
74.0
85.7
4400.7
6
7
67.4
61.7
58.1
56.7
57.6
61.0
67.0
75.8
87.6
02.6
7
8
69.0
63.3
59.7
58.3
59.3
62.7
68.8
77.7
89.5
04.6
8
9
70.5
64.9
61.3
60.0
61.0
64.5
70.6
79.5
91.4
06.5
9
10
3472.1
3566.5
3663.0
3761.7
3862.7
3966.2
4072.4
4181.3
4293.3
4408.5
10
11
73.6
68.1
64.6
63.3
64.4
68.0
74.2
83.2
95.2
10.4
11
12
75.2
69.7
66.2
65.0
66.1
69.7
76.0
85.0
97.1
12.4
12
13
76.7
71.3
67.9
66.7
67.8
71.5
77.7
86.9
99.0
14.3
13
14
78.3
72.8
69.5
68.3
69.5
73.2
79.5
88.7
4300.9
16.3
14
15
3479.9
3574.4
3671.1
3770.0
3871.2
3975.0
4081.3
4190.6
4302.8
4418.2
15
16
81.4
76.0
72.7
71.7
72.9
76.7
83.1
92.4
04.7
20.2
16
17
83.0
77.6
74.4
73.3
74.6
78.5
84.9
94.2
06.6
22.1
17
18
84.5
79.2
76.0
75.0
76.3
80.2
86.7
96.1
08.5
24.1
18
19
86.1
80.8
77.6
76.7
78.1
82.0
88.5
97.9
10.4
26.1
19
20
3487.7
3582.4
3679.3
3778.3
3879.8
3983.7
4090.3
4199.8
4312.3
4428.0
20
21
89.2
84.0
80.9
80.0
81.5
85.5
92.1
4201.6
14.2
30.0
21
22
90.8
85.6
82.5
81.7
83.2
87.2
93.9
03.5
16.1
31.9
22
23
92.4
87.2
84.2
83.3
84.9
89.0
95.7
05.3
18.0
33.9
23
24
93.9
88.8
85.8
85.0
86.6
90.7
97.5
07.2
19.9
35.8
24
25
3495.5
3590.4
3687.4
3786.7
3888.3
3992.5
4099.3
4209.0
4321.8
4437.8
25
26
97.1
92.0
89.1
88.4
90.0
94.3
4101.1
10.9
23.7
39.8
26
27
98.6
93.6
90.7
90.0
91.8
96.0
02.9
12.8
25.6
41.7
27
28
3500.2
95.2
92.3
91.7
93.5
97.8
04.8
14.6
27.5
43.7
28
29
01.8
96.8
94.0
93.4
95.2
99.5
06.6
16.5
29.4
45.7
29
30
3503.3
3598.4
3695.6
3795.1
3896.9
4001.3
4108.4
4218.3
4331.3
4447.6
30
31
04.9
3600.0
97.3
96.8
98.6
03.1
10.2
20.2
33.2
49.6
31
32
06.5
01.6
98.9
98.4
3900.4
04.8
12.0
22.0
35.2
51.6
32
33
08.0
03.2
3700.5
3800.1
02.1
06.6
13.8
23.9
37.1
53.5
33
34
09.6
04.8
02.2
01.8
03.8
08.3
15.6
25.8
39.0
55.5
34
35
3511.2
3606.4
3703.8
3803.5
3905.5
4010.1
4117.4
4227.6
4340.9
4457.5
35
36
12.7
08.0
05.5
05.1
07.2
11.9
19.2
29.5
42.8
59.4
36
37
14.3
09.6
07.1
06.8
09.0
13.6
21.0
31.3
44.7
61.4
37
38
15.9
11.2
08.7
08.5
10.7
15.4
22.9
33.2
46.6
63.4
38
39
17.5
12.8
10.4
10.2
12.4
17.2
24.7
35.1
48.6
65.4
39
40
3519.0
3614.5
3712.0
3811.9
3914.1
4018.9
4126.5
4236.9
4350.5
4467.3
40
41
20.6
16.1
13.7
13.6
15.9
20.7
28.3
38.8
52.4
69.3
41
42
22.2
17.7
15.3
15.2
17.6
22.5
30.1
40.7
54.3
71.3
42
43
23.7
19.3
17.0
17.0
19.3
24.3
31.9
42.5
56.2
73.3
43
44
25.3
20.9
18.6
18.6
21.0
26.0
33.8
44.4
58.2
75.3
44
45
3526.9
3622.5
3720.3
3820.3
3922.8
4027.8
4135.6
4246.3
4360.1
4477.2
45
46
28.5
24.1
21.9
22.0
24.5
29.6
37.4
48.1
62.0
79.2
46
47
30.1
25.7
23.6
23.7
26.2
31.4
39.2
50.0
63.9
81.2
47
48
31.6
27.3
25.2
25.4
28.0
33.1
41.0
51.9
65.9
83.2
48
49
33.2
29.0
26.9
27.1
29.7
34.9
42.9
53.8
67.8
85.2
49
50
3534.8
3630.6
3728.5
3828.7
3931.4
4036.7
4144.7
4255.6
4369.7
4487.2
50
51
36.4
32.2
30.2
30.4
33.2
38.5
46.5
57.5
71.7
89.1
51
52
37.9
33.8
31.8
32.1
34.9
40.2
48.3
59.4
73.6
91.1
52
53
39.5
35.4
33.5
33.8
36.6
42.0
50.2
61.3
75.5
93.1
53
54
41.1
37.0
35.1
35.5
38.4
43.8
52.0
63.1
77.4
95.1
54
55
3542.7
3638.6
3736.8
3837.2
3940.1
4045.6
4153.8
4265.0
4379.4
4497.1
55
56
44.3
40.3
38.4
38.9
41.8
47.4
55.7
66.9
81.3
99.1
56
57
45.9
41.9
40.1
40.6
43.6
49.1
57.5
68.8
83.2
4501.1
57
58
47.4
43.5
41.7
42.3
45.3
50.9
59.3
70.7
85.2
03.1
58
59
49.0
45.1
43.4
45.0
47.0
52.7
61.1
72.5
87.1
05.1
59
60
3550.6
3646.7
3745.1
3845.7
3948.8
4054.5
4163.0
4274.4
4389.1
4507.1
60
/
50°
51°
52°
53°
54°
55°
56°
57°
58°
59°
Table 6
Table 7 247
Combined Correction for Observed
Sextant Altitudes
Correction for Dip of
Sea Horizon
(Sun or Star)
OBSEHVED
ALTITUDE
CORRECTION
For Sun (to
be added to
observed alti-
tude)
For Star (to
be subtracted
from observed
altitude)
5°
6' 14"
9' 55"
6
7 41
8 28
7
8 45
7 24
8
9 35
6 34
9
10 16
5 53
10
10 50
5 19
11
11 17
4 51
12
11 41
4 27
13
12 2
4 7
14
12 19
3 49
15
12 34
3 34
20
13 29
2 39
25
14 3
2 5
30
14 26
1 41
35
14 44
1 23
40
14 57
1 10
45
15 8
0 58
50
15 17
0 49
55
15 25
0 40
60
15 31
0 34
65
15 37
0 27 .
70
15 42
0 21
75
15 47
0 16
80
15 52
0 10
85
15 55
0 5
HEIGHT OP
OBSERVER'S
EYE ABOVE
SEA LEVEL
(feet)
DIP CORREC-
TION (to be
subtracted
from
observed
altitude)
4
1' 58"
6
2 24
8
2 46
10
3 06
12
3 24
14
3 40
16
3 55
18
4 9
20
4 23
22
4 36
24
4 48
26
5 0
28
5 11
30
5 22
35
5 48
40
6 12
45
6 36
50
6 56
55
7 16
60
7 35
70
8 12
85
9 2
100
9 48
Small supplementary correction, for Sun
only.
Jan. to March \ jj int.
and Oct. to Dec. ;add 10 "•
April to Sept., subtract 10".
The dip correction is not
required when the artificial
horizon is used.
248
Table 8
To Change Hours and Minutes into Decimals of a Day
HOURS EXPRESSED
AS DECIMAL PARTS
OF A DAY
HOURS
DECIMAL
1
.0416
2
.0833
3
.1250
4
.1666
5
.2083
6
.2500
7
.2916
8
.3333
9
.3750
10
.4166
11
.4583
12
.5000
13
.5416
14
.5833
15
.6249
16
.6666
17
.7083
18
.7500
19
.7916
20
.8333
21
.8749
22
.9166
23
.9583
24
1.0000
MINUTES EXPRESSED AS DECIMAL PARTS
OF A DAY
MINUTES
DECIMAL
MINUTES
DECIMAL
1
.0006
31
.0215
2
.0013
32
.0222
3
.0020
33
.0229
4
.0027
34
.0236
5
.0034
35
.0243
6
.0041
36
.0250
7
.0048
37
.0256
8
.0055
38
.0263
9
.0062
39
.0270
10
.0069
40
.0277
11
.0076
41
.0284
12
.0083
42
.0291
13
.0090
43
.0298
14
.0097
44
.0305
15
.0104
45
.0312
16
.0111
46
.0319
17
.0118
47
.0326
18
.0125
48
.0333
19
.0131
49
.0340
20
.0138
50
.0347
21
.0145
51
.0354
22
.0152
52
.0361
23
.0159
53
.0368
24
.0166
54
.0375
25
.0173
55
.0381
26
.0180
56
.0388
27
.0187
57
.0395
28
.0194
58
.0402
29
.0201
59
.0409
30
.0208
bO
.0416
Table 9
249
To Interchange Degrees and Minutes of Longitude and Hours, Minutes,
and Seconds of Time. Part 1
0*
1*
2*
3*
4A
6*
6*
7*
8*
9*
10*
11*
om
0°
15°
30°
45°
60°
75°
90°
105°
120°
135°
150°
165°
4
1
16
31
46
61
76
91
106
121
136
151
166
8
2
17
32
47
62
77
92
107
122
137
152
167
12
3
18
33
48
63
78
93
108
123
138
153
168
16
4
19
34
49
64
79
94
109
124
139
154
169
20
5
20
35
50
65
80
95
110
125
140
155
170
24
6
21
36
51
66
81
96
111
126
141
156
171
28
7
22
37
52
67
82
97
112
127
142
157
172
32
8
23
38
53
68
83
98
113
128
143
158
173
36
9
24
39
54
69
84
99
114
129
144
159
174
40
10
25
40
55
70
85
100
115
130
145
160
175
44
11
26
41
56
71
86
101
116
131
146
161
176
48
12
27
42
57
72
87
102
117
132
147
162
177
52
13
28
43
58
73
88
103
118
133
148
163
178
56
14
29
44
59
74
89
104
119
134
149
164
179
12*
13*
14*
f5*
16*
17*
18*
19*
20*
21*
22*
23*
0"!
180°
195°
210°
225°
240°
255°
270°
285°
300°
315°
330°
345°
4
181
196
211
226
241
256
271
286
301
316
331
346
8
182
197
212
227
242
257
272
287
302
317
332
347
12
183
198
213
228
243
258
273
288
303
318
333
348
16
184
199
214
229
244
259
274
289
304
319
334
349
20
185
200
215
230
245
260
275
290
305
320
335
350
24
186
201
216
231
246
261
276
291
306
321
336
351
28
187
202
217
232
247
262
277
292
307
322
337
352
32
188
203
218
233
248
263
278
293
308
323
338
353
36
189
204
219
234
249
264
279
294
309
324
339
354
40
190
205
220
235
250
265
280
295
310
325
340
355
44
191
206
221
236
251
266
281
296
311
326
341
356
48
192
207
222
237
252
267
282
297
312
327
342
357
52
193
208
223
238
253
268
283
298
313
328
343
358
56
194
209
224
239
254
269
284
299
314
329
344
359
Part 2
EXPLANATION OP TABLE 9
1. To change degrees of longitude into hours and
minutes of time : Find the number of degrees in Part 1.
The required hours will then be found at the head of the
column containing the degrees, and the required min-
utes at the left-hand end of the line containing the
degrees.
Examples: 113° = 7* 32m ; 294° = 19* 36m.
2. To change minutes of longitude into minutes and
seconds of time : Find the minutes of longitude in Part 2.
The required minutes and seconds of time will again
be found at the head of the column and the left-hand end
of the line.
Examples : 43' = 2m 52s ; 28' = lm 52".
3. 1 and 2 can be combined by addition.
Examples : 113° 43' = 7* 34m 52s.
294° 28' = 19* 37m 52».
4. To change hours and minutes of time into degrees
and minutes of longitude : Find the number of hours at
the head of one of the columns of Part 1 ; then run down
the column until you reach a line having at its left-hand
end a number of minutes equal to (or just smaller than)
the given number of minutes of time. Where that line
and column meet you will find the required degrees of longitude.
Examples: 7'« 32m = 113°; 19* 36m = 294°.
5. To change minutes and seconds of time into minutes of longitude : Find the number of
minutes of time at the head of one of the columns of Part 2 ; then run down the column until
you reach a line having at its left-hand end a number of seconds equal (or nearly equal) to
the given number of seconds of time. Where that line and column meet you will find the
minutes of longitude.
Examples : 2m 52* = 43' ; lm 52s = 28'.
6. 4 and 5 can be combined by addition :
Examples : 7» 34m 52' = 1 13° 43' ; 19* 37m 52* = 294° 28'.
Qm
1"'
2m
8»
0s
0'
15'
30'
45'
4
1
16
31
46
8
2
17
32
47
12
3
18
33
48
16
4
19
34
49
20
5
20
35
50
24
6
21
36
51
28
7
22
37
52
32
8
23
38
53
36
9
24
39
54
40
10
25
40
55
44
11
26
41
56
48
12
27
42
57
52
13
28
43
58
56
14
29
44
59
250
Table 10. Haversine Table
s '
OhOm 0°
Oh 4™ 1°
Oh s™ 2°
Qh 1Sm 30
1I.IV.
No.
Hav.
No.
Hav.
No.
Bar.
No.
0 0
0.00000
5.88168
0.00008
6.48371
0.00030
6.83584
0.00069
4 1
2.32539
.00000
.89604
.00008
.49092
.00031
.84065
.00069
8 2
.92745
.00000
.91016
.00008
.49807
.00031
.84543
.00070
12 3
3.27963
.00000
.92406
.00008
.50516
.00032
.85019
.00071
16 4
.52951
.00000
.93774
.00009
.51219
.00033
.85492
.00072
20 5
3.72333
0.00000
5.95121
0.00009
6.51916
0.00033
6.85963
0.00072
24 6
.88169
.00000
.96447
.00009
.52608
.00034
.86431
.00073
28 7
4.01559
.00000
.97753
.00010
.53295
.00034
.86897
.00074
32 8
.13157
.00000
.99040
.00010
.53976
.00035
.87360
.00075
36 9
.23388
.00000
6.00308
.00010
.54652
.00035
.87821
.00076
40 10
4.32539
0.00000
6.01557
0.00010
6.55323
0.00036
6.88279
0.00076
44 11
.40818
.00000
.02789
.00011
.55988
.00036
.88735
.00077
48 12
.48375
.00000
.04004
.00011
.56649
.00037
.89188
.00078
52 13
.55328
.00000
.05202
.00011
.57304
.00037
.89639
.00079
56 14
.61765
.00000
.06384
.00012
.57955
.00038
.90088
.00080
s '
Qh jm QO
Oh 6>n jo
Qhgm 2°
Oh 13m 3°
0 15
4.67757
0.00000
6.07550
0.00012
6.58600
0.00039
6.90535
0.00080
4 16
.73363
.00001
.08700
.00012
.59241
.00039
.90979
.00081
S 17
.78629
.00001
.09836
.00013
.59878
.00040
.91421
.00082
12 18
.83594
.00001
.10956
.00013
.60509
.00040
.91860
.00083
76 19
.88290
.00001
.12063
.00013
.61136
.00041
.92298
.00084
20 20
4.92745
0.00001
6.13155
0.00014
6.61759
0.00041
6.92733
0.00085
24 21
.96983
.00001
.14234
.00014
.62377
.00042
.93166
.00085
2S 22
5.01024
.00001
.15300
.00014
.62991
.00043
.93597
.00086
32 23
.04885
.00001
.16353
.00015
.63600
.00043
.94026
.00087
36 24
.08581
.00001
.17393
.00015
.64205
.00044
.94453
.00088
40 25
5.12127
0.00001
6.18421
0.00015
6.64806
0.00044
6.94877
0.00089
44 26
.15534
.00001
.19437
.00016
.65403
.00045
.95300
.00090
45 27
.18812
.00002
.20441
.00016
.65996
.00046
.95720
.00091
52 28
.21971
.00002
.21433
.00016
.66585
.00046
.96139
.00091
56 29
.25019
.00002
.22415
.00017
.67170
.00047
.96555
.00092
s '
Qh 2m QO
Qhffm jo
Oh iom 2°
Oh 14™ 3°
0- 30
5.27963
0.00002
6.23385
0.00017
6.67751
0.00048
6.96970
0.00093
4 31
.30811
.00002
.24345
.00018
.68328
.00048
.97382
.00094
8 32
.33569
.00002
.25294
.00018
.68901
.00049
.97793
.00095
72 33
.36242
.00002
.26233
.00018
.69470
.00050
.98201
.00096
76 34
.38835
.00002
.27162
.00019
.70036
.00050
.98608
.00097
20 35
5.41352
0.00003
6.28081
0.00019
6.70598
0.00051
6.99013
0.00098
24 36
.43799
.00003
.28991
.00019
.71157
.00051
.99416
.00099
28 37
.46179
.00003
.29891
.00020
.71712
.00052
.99817
.00100
32 38
.48496
.00003
.30781
.00020
.72263
.00053
7.00216
.00101
3(5 39
.50752
.00003
.31663
.00021
.72811
.00053
.00613
.00101
40 40
5.52951
0.00003
6.32536
0.00021
6.73355
0.00054
7.01009
0.00102
44 41
.55095
.00004
.33400
.00022
.73896
.00055
.01403
.00103
48 42
.57189
.00004
.34256
.00022
.74434
.00056
.01795
.00104
52 43
.59232
.00004
.35103
.00022
.74969
.00056
.02185
.00105
56 44
.61229
.00004
.35943
.00023
.75500
.00057
02573
.00106
s '
Qh 3m 00
Qh 7m JO
0*11™ 2°
Qh 15m 3°
0 45
5.63181
0.00004
6.36774
0.00023
6.76028
0.00058
7.02960
0.00107
4 46
.65090
.00004
.37597
.00024
.76552
.00058
.03345
.00108
5 47
.66958
.00005
.38412
.00024
.77074
.00059
.03729
.00109
/2 48
.68787
.00005
.39220
.00025
.77592
.00060
.04110
.00110
16 49
.70578
.00005
.40021
.00025
.78108
.00060
.04490
.00111
20 50
5.72332
0.00005
6.40814
0.00026
6.78620
0.00061
7.04869
0.00112
24 51
.74052
.00006
.41600
.00026
.79129
.00062
.05245
.00113
28 52
.75739
.00006
.42379
.00027
.79630
.00063
.05620
.00114
32 53
.77394
.00006
.43151
.00027
.80139
.00063
.05994
.00115
36 54
.79017
.00006
.43916
.00027
.80640
.00064
.06366
.00116
40 55
5.80611
0.00006
6.44675
0.00028
6.81137
0.00065
7.06736
0.00117
44 56
.82176
.00007
.45427
.00028
.81632
.00066
.07105
.00118
45 57
.83713
.00007
.46172
.00029
.82124
.00066
.07472
.00119
52 58
.85224
.00007
.46911
.00029
.82614
.00067
.07837
.00120
56 59
.86709
.00007
.47644
.00030
.83100
.00068
.08201
.00121
<?0 60
5.88168
0.00008
6.48371
0.00030
6.83584
0.00069
7.08564
0.00122
Table 10. Hayersine Table
251
S '
0" 16™ 4°
0* 20m 5°
0A 24m 6° .
0* 28™ 7°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
7.08564
0.00122
7.27936
0.00190
7.43760
0.00274
7.57135
0.00373
4 l
.08925
.00123
.28225
.00192
.44001
.00275
.57341
.00374 |
8 2
.09284
.00124
.28513
.00193
.44241
.00277
.57547
.00376
12 3
.09642
.00125
.28800
.00194
.44480
.00278
.57752
.00378
16 4
.09999
.00126
.29086
.00195
.44719
.00280
.57957
.00380
20 5
7.10354
0.00127
7.29371
0.00197
7.44957
0.00282
7.58162
0.00382
24 6
.10708
.00128
.29655
.00198
.45194
.00283
.58366
.00383
28 7
.11060
.00129
.29938
.00199
.45431
.00285
.58569
.00385
32 8
.11411
.00130
.30220
.00201
.45667
.00286
.58772
.00387
36 9
.11760
.00131
.30502
.00202
.45903
.00288
.58974
.00389
40 10
7.12108
0.00132
7.30782
0.00203
7.46138
0.00289
7.59176
0.00391
44 11
.12455
.00133
.31062
.00204
.46372
.00291
.59378
.00392
48 12
.12800
.00134
.31340
.00206
.46605
.00292
.59579
.00394
52 13
.13144
.00135
.31618
.00207
.46838
.00294
.59779
.00396
56 14
.13486
.00136
.31895
.00208
.47071
.00296
.59979
.00398
s '
Qh 17m 4°
Qh 2/m 5°
0* 25™ 6°
Oh 29™ 7°
0 15
7.13827
0.00137
7.32171
0.00210
7.47302
0.00297
7.60179
0.00400
4 16
.14167
.00139
.32446
.00211
.47533
.00299
.60378
.00402
5 17
.14506
.00140
.32720
.00212
.47764
.00300
.60577
.00403
12 18
.14843
.00141
.32994
.00214
.47994
.00302
.60775
.00405
1(5 19
.15179
.00142
.33266
.00215
.48223
.00304
.60973
.00407
£0 20
7.15513
0.00143
7.33538
0.00216
7.48452
0.00305
7.61170
0.00409
24 21
.15846
.00144
.33809
.00218
.48680
.00307
.61367
.00411
£5 22
.16178
.00145
.34079
.00219
.48907
.00308
.61564
.00413
32 23
.16509
.00146
.34348
.00221
.49134
.00310
.61760
.00415
36 24
.16839
.00147
.34616
.00222
.49360
.00312
.61955
.00416
40 25
7.17167
0.00148
7.34884
0.00223
7.49586
0.00313
7.62151
0.00418
44 26
.17494
.00150
.35150
.00225
.49811
.00315
.62345
.00420
45 27
.17820
.00151
.35416
.00226
.50036
.00316
.62540
.00422
52 28
.18144
.00152
.35681
.00227
.50259
.00318
.62733
.00424
55 29
.18468
.00153
.35945
.00229
.50483
.00320
.62927
.00426
s '
0* 18™ 4°
0*22™ 5°
Oh 26™ 6°
0*30™ 7°
0 30
7.18790
0.00154
7.36209
0.00230
7.50706
0.00321
7.63120
0.00428
4 31
.19111
.00155
.36471
.00232
.50928
.00323
.63312
.00430
8 32
.19430
.00156
.36733
.00233
.51149
.00325
.63504
.00432
.72 33
.19749
.00158
.36994
.00234
.51370
.00326
.63696
.00433
/'/ 34
.20066
.00159
.37254
.00236
.51591
.00328
.63887
.00435
20 35
7.20383
0.00160
7.37514
0.00237
7.51811
0.00330
7.64078
0.00437
24 36
.20698
.00161
.37773
.00239
.52030
.00331
.64269
.00439
28 37
.21012
.00162
.38030
.00240
.52249
.00333
.64458
.00441
32 38
.21325
.00163
.38288
.00241
.52467
.00335
.64648
.00443
36 39
.21636
.00165
.38544
.00243
.52685
.00336
.64837
.00445
40 40
7.21947
0.00166
7.38800
0.00244
7.52902
0.00338
7.65026
0.00447
44 41
.22256
.00167
.39054
.00246
.53119
.00340
.65214
.00449
48 42
.22565
.00168
.39309
.00247
.53335
.00341
.65402
.00451
52 43
.22872
.00169
.39562
.00249
.53550
.00343
.65590
.00453
56 44
.23178
.00171
.39815
.00250
.53766
.00345
.65777
.00455
s '
0* 1ST 4°
0*23™ 5°
Oh 27m 6°
Oh sim 7°
0 45
7.23483
0.00172
7.40067
0.00252
7.53980
0.00347
7.65964
0.00457
4 46
.23787
.00173
.40318
.00253
.54194
.00348
.66150
.00459
S 47
.24090
.00174
.40568
.00255
.54407
.00350
.66336
.00461
J2 48
.24392
.00175
.40818
.00256
.54620
.00352
.66521
.00463
16 49
.24693
.00177
.41067
.00257
.54833
.00353
.66706
.00465
20 50
7.24993
0.00178
7.41315
0.00259
7.55045
0.00355
7.66891
0.00467
24 51
.25292
.00179
.41563
.00260
.55256
.00357
.67075
00469
2S 52
.25590
.00180
.41810
.00262
.55467
.00359
.67259
.00471
{32 53
.25886
.00181
.42056
.00263
.55677
.00360
.67443
.00473
36 54
.26182
.00183
.42301
.00265
.55887
.00362
.67626
.00475
40 55
7.26477
0.00184
7.42546
0.00266
7.56096
0.00364
7.67809
0.00477
44 56
.26771
.00185
.42790
.00268
.56305
.00366
.67991
.00479
48 57
.27064
.00186
.43034
.00269
.56513
.00367
.68173
.00481
52 58
.27355
.00188
.43277
.00271
.56721
.00369
.68355
.00483
56 59
.27646
.00189
.43519
.00272
.56928
.00371
.68536
.00485
60 60
7.27936
0.00190
7.43760
0.00274
7.57135
0.00373
7.68717
0.00487
252
Table 10. Haversine Table
s '
Oh32m 8°
Oh 36™ 9°
Qh 40m 10°
Oh 44m 11°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
7.68717
0.00487
7.78929
0.00616
7.88059
0.00760
7.96315
0.00919
4 1
.68897
.00489
.79089
.00618
.88203
.00762
.96446
.00921
8 2
.69077
.00491
.79249
.00620
.88348
.00765
.96577
.00924
12 3
.69257
.00493
.79409
.00622
.88491
.00767
.96707
.00927
16 4
.69437
.00495
.79568
.00625
.88635
.00770
.96838
.00930
20 5
7.69616
0.00497
7.79728
0.00627
7.88778
0.00772
7.96968
0.00933
24 6
.69794
.00499
.79886
.00629
.88921
.00775
.97098
.00935
28 7
.69972
.00501
.80045
.00632
.89064
.00777
.97228
.00938
32 8
.70150
.00503
.80203
.00634
.89207
.00780
.97358
.00941
36 9
.70328
.00505
.80361
.00636
.89349
.00783
.97478
.00944
40 10
7.70505
0.00507
7.80519
0.00639
7.89491
0.00785
7.97617
0.00947
44 11
.70682
.00509
.80677
.00641
.89633
.00788
.97746
.00949
48 12
.70858
.00511
.80834
.00643
.89775
.00790
.97875
00952
52 13
.71034
.00513
.80991
.00646
.89916
.00793
.98003
.00955
f>6 14
.71210
.00515
.81147
.00648
.90057
.00795
.98132
.00958
s '
Qh 33m 8°
Oh 37m 90
Qh 41™ 10°
0^ 45m 11°
f 15
7.71385
0.00517
7.81303
0.00650
7.90198
0.00798
7.98260
0.00961
4 16
.71560
.00520
.81459
.00653
.90339
.00801
.98389
.00964
/? 17
.71735
.00522
.81615
.00655
.90480
.00803
.98517
.00966
12 18
.71909
.00524
.81771
.00657
.90620
.00806
.98644
.00969
/6 19
.72083
.00526
,81926
.00660
.90760
.00808
.98772
.00972
£0 20
7.72257
0.00528
7.82081
0.00662
7.90900
0.00811
7.98899
0.00975
24 21
.72430
.00530
.82235
.00664
.91039
.00814
.99027
.C0978
2S 22
.72603
.00532
.82390
.00667
.91179
.00816
.99154
.00981
32 23
.72775
.00534
.82544
.00669
.91318
.00819
.99281
.00984
3<? 24
.72948
.00536
.82698
.00671
.91457
.00821
.99407
.00986
40 25
7.73119
0.00539
7.82851
0.00674
7.91596
0.00824
7.99534
0.00989
44 26
.73291
.00541
.83004
.00676
.91734
.00827
.99660
.00992
45 27
.73462
.00543
.83157
.00679
.91872
.00829
.99786
.00995
52 28
.73633
.00545
.83310
.00681
.92010
.00832
.99912
.00998
5<J 29
.73803
.00547
.83463
.00683
.92148
.00835
8.00038
.01001
s '
Oh 34m 8°
Oh $8™ 9°
0*42™ 10°
0* 46™ 11°
0 30
7.73974
0.00549
7.83615
0.00686
7.92286
0.00837
8.00163
0.01004
4 31
.74143
.00551
.83767
.00688
.92423
.00840
.00289
.01007
8 32
.74313
.00554
.83918
.00691
.92560
.00843
.00414
.01010
J2 33
.74482
.00556
.84070
.00693
.92697
.00845
.00539
.01012
itf 34
.74651
.00558
.84221
.00695
.92834
.00848
.00664
.01015
20 35
7.74819
0.00560
7.84372
0.00698
7.92970
0.00851
8.00788
0.01018
24 36
.74988
.00562
.84522
.00700
.93107
.00853
.00913
.01021
28 37
.75155
.00564
.84672
.00703
.93243
.00856
.01037
.01024
32 38
.75323
.00567
.84822
.00705
.93379
.00859
.01161
.01027
3<S 39-
.75490
.00569
.84972
.00707
.93514
.00861
.01285
.01030
40 40
7.75657
0.00571
7.85122
0.00710
7.93650
0.00864
8.01409
0.01033
44 41
.75824
.00573
.85271
.00712
.93785
.00867
.01532
.01036
48 42
.75990
.00575
.85420
.00715
.93920
.00869
.01656
.01039
52 43
.76156
.00578
.85569
.00717
.94055
.00872
.01779
.01042
50 44
.76321
.00580
.85717
.00720
.94189
.00875
.01902
.01045
s '
Oh 35m 8°
0*35™ 9°
Oh 43™ 10°
Oh 47m 11°
0 45
7.76487
0.00582
7.85866
0.00722
7.94324
0.00877
8.02025
0.01048
4 46
.76652
.00584
.86014
.00725
.94458
.00880
.02148
.01051
S 47
.76816
.00586
.86161
.00727
.94592
.00883
.02270
.01054
/.' 48
.76981
.00589
.86309
.00730
.94726
.00886
.02392
.01057
16 49
.77145
.00591
.86456
.00732
.94859
.00888
.02515
.01060
20 50
7.77308
0.00593
7.86603
0.00735
7.94992
0.00891
8.02637
0.01063
24 51
.77472
.00595
.86750
.00737
.95126
.00894
.02758
.01066
25 52
.77635
.00598
.86896
.00740
.95259
.00897
.02880
.01069
32 53
.77798
.00600
.87042
.00742
.95391
.00899
.03001
.01072
36 54
.77960
.00602
.87188
.00745
.95524
.00902
.03123
.01075
40 55
7.78122
0.00604
7.87334
0.00747
7.95656
0.00905
8.03244
0.01078
44 56
.78284
.00607
.87480
.00750
.95788
.00908
.03365
.01081
45 57
.78446
.00609
.87625
.00752
.95920
.00910
.03486
.01084
52 58
.78607
.00611
.87770
.00755
.96052
.00913
.03606
.01087
56 59
.78768
.00613
.87915
.00757
.96183
.00916
.03727
.01090
60 60
7.78929
0.00616
7.88059
0.00760
7.96315
0.00919
8.03847
001093
Table 10. Haversine Table
253
s '
0* 4#"' 12°
0A 52m 13°
0A 56™ 14°
lh (jm. 150
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
8.03847
0.01093
8.10772
0.01282
8.17179
0.01485
8.23140
0.01704
4 1
.03967
.01096
.10883
.01285
.17282
.01489
.23235
.01707
8 2
.04087
.01099
.10993
.01288
.17384
.01492
.23331
.01711
12 3
.04207
.01102
.11104
.01291
.17487
.01496
.23427
.01715
16 4
.04326
.01105
.11214
.01295
.17590
.01499
.23523
.01719
20 5
8.04446
0.01108
8.11324
0.01298
8.17692
0.01503
8.23618
0.01723
24 6
.04505
.01111
.11435
.01301
.17794
.01506
.23713
.01726
28 1
.04684
.01114
.11544
.01305
.17896
.01510
.23809
.01730
3> 8
.04803
.01117
.11654
.01308
.17998
.01513
.23904
.01734
36 9
.04922
.01120
.11764
.01311
.18100
.01517
.23999
.01738
40 10
8.05041
0.01123
8.11873
0.01314
8.18202
0.01521
8.24094
0.01742
44 11
.05159
.01126
.11983
.01317
.18303
.01524
.24189
.01745
48 12
.05277
.01129
.12092
.01321
.18405
.01528
.24283
.01749
52 13
.05395
.01132
.12201
.01324
.18506
.01531
.24378
.01753
56 14
.05513
.01135
.12310
.01328
.18607
.01535
.24473
.01757
s '
0* A9m 12°
0ft 53"1 13°
Oh 57m 14°
!h jm 15°
0 15
8.05631
0.01138
8.12419
0.01331
8.18709
0.01538
8.24567
0.01761
4 16
.05749
.01142
.12528
.01334
.18810
.01542
.24661
.01764
S 17
.05866
.01145
.12636
.01338
.18910
.01546
.24755
.01768
12 18
.05984
.01148
.12745
.01341
.19011
.01549
.24850
.01772
10 19
.06101
.01151
.12853
.01344
.19112
.01553
.24944
.01776
20 20
8.06218
0.01154
8.12961
0.01348
8.19212
0.01556
8.25037
0.01780
24 21
.06335
.01157
.13069
.01351
.19313
.01560
.25131
.01784
2S 22
.06451
.01160
.13177
.01354
.19413
.01564
.25225
.01788
32 23
.06568
.01163
.13285
.01358
.19513
.01567
.25319
.01791
36 24
.06684
.01166
.13392
.01361
.19613
.01571
.25412
.01795
40 25
8.06800
0.01170
8.13500
0.01365
8.19713
0.01574
8.25505
0.01799
44 26
.06917
.01173
.13607
.01368
.19813
.01578
.25599
.01803
45 27
.07032
.01176
.13714
.01371
.19913
.01582
.25692
.01807
52 28
.07148
.01179
.13822
.01375
.20012
.01585
.25785
.01811
5(5 29
.07264
.01182
.13928
.01378
.20112
.01589
.25878
.01815
s '
0* 50" 12°
0* 54"* 13°
0* 68™ 14°
lh §m 15°
0 30
8.07379
0.01185
8.14035
0.01382
8.20211
0.01593
8.25971
0.01818
4 31
.07494
.01188
.14142
.01385
.20310
.01596
.26064
.01822
8 32
.07610
.01192
.14248
.01388
.20410
.01600
.26156
.01826
J2 33
.07725
.01195
.14355
.01392
.20509
.01604
.26249
.01830
/6 34
.07839
.01198
.14461
.01395
.20608
.01607
.26341
.01834
20 35
8.07954
0.01201
8.14567
0.01399
8.20706
0.01611
8.26434
0.01838
24 36
.08069
.01204
.14673
.01402
.20805
.01615
.26526
.01842
28 37
.08183
.01207
.14779
.01405
.20904
.01618
.26618
.01846
32 38
.08297
.01211
.14885
.01409
.21002
.01622
.26710
.01850
36 39
.08411
.01214
.14991
.01412
.21100
.01626
.26802
.01854
40 40
8.08525
0.01217
8.15096
0.01416
8.21199
0.01629
8.26894
0.01858
4-4 41
. .08639
.01220
.15201
.01419
.21297
.01633
.26986
.01861
48 42
.08752
.01223
.15307
.01423
.21395
.01637
.27078
.01865
52 43
.08866
.01226
.15412
.01426
.21493
.01640
.27169
.01869
56 44
.08979
.01230
.15517
.01429
.21590
.01644
.27261
.01873
8 '
0* 51m 12°
0A 55m 13°
0* 59m 14°
lh S™ 15°
0 45
8.09092
0.01233
8.15622
0.01433
8.21688
0.01648
8.27352
0.01877
4 46
.09205
.01236
.15726
.01436
.21785
.01651
.27443
.01881
5 47
.09318
.01239
.15831
.01440
.21883
.01655
.27534
.01885
/# 48
.09431
.01243
.15935
.01443
.21980
.01659
.27626
.01889
16 49
.09543
.01246
.16040
.01447
.22077
.01663
.27717
.01893
20 50
8.09656
0.01249
8.16144
0.01450
8.22175
0.01666
8.27807
0.01897
24 51.
.09768
.01252
.16248
.01454
.22272
.01670
.27898
.01901
2S 52
.09880
.01255
.16352
.01457
.22368
.01674
.27989
.01905
32 53
.09992
.01259
.16456
.01461
.22465
.01677
.28080
.01909
36 54
.10104
.01262
.16559
.01464
.22562
.01681
.28170
.01913
40 55
8.10216
0.01265
8.16663
0.01468
8.22658
0.01685
8.28260
0.01917
44 56
.10327
.01268
.16766
.01471
.22755
.01689
.28351
.01921
48 57
.10439
.01272
.16870
.01475
.22851
.01692
.28441
.01925
52 58
.10550
.01275
.16973
.01478
.22947
.01696
.28531
.01929
56 59
.10661
.01278
.17076
.01482
.23044
.01700
.28621
.01933
60 60
8.10772
0.01282
8.17179
0.01485
8.23140
0.01704
8.28711
0.01937
254
Table 10. Haversine Table
s '
1* 4m 16°
Ik 8m 17°
Ik 12™ 18°
lh 16™ 19°
Hav.
No.
Hav.
No.
Uav.
No.
Hav.
No.
0 0
8.28711
0.01937
8.33940
0.02185
8.38867
0.02447
8.43522
0.02724
4 1
.28801
.01941
.34025
.02189
.38946
.02452
.43597
.02729
8 2
.28891
.01945
.34109
.02193
.39026
.02456
.43673
.02734
13 3
.28980
.01949
.34194
.02198
.39105
.02461
.43748
.02738
16 4
.29070
.01953
.34278
.02202
.39185
.02465
.43823
.02743
20 5
8.29159
0.01957
8.34362
0.02206
8.39264
0.02470
8.43899
0.02748
24 6
.29249
.01961
.34446
.02210
.39344
.02474
.43974
.02753
28 7
.29338
.01965
.34530
.02215
.39423
.02479
.44049
.02757
32 8
.29427
.01969
.34614
.02219
.39502
.02483
.44124
.02762
36 9
.29516
.01973
.34698
.02223
.39581
.02488
.44199
.02767
40 10
8.29605
0.01977
8.34782
0.02227
8.39660
0.02492
8.44273
0.02772
44 11
.29694
.01981
.34865
.02232
.39739
.02497
.44348
.02776
48 12
.29783
.01985
.34949
.02236
.39818
.02501
.44423
.02781
52 13
.29872
.01989
.35032
.02240
.39897
.02506
.44498
.02786
56 14
.29960
.01993
.35116
.02245
.39976
.02510
.44572
.02791
s '
!h 6m 16o
lh gm 17°
Ik 13™ 18°
Ik 17m 19°
0 15
8.30049
0.01998
8.35199
0.02249
8.40055
0.02515
8.44647
0.02796
4 16
.30137
.02002
.35282
.02253
.40133
.02520
.44721
.02800
S 17
.30226
.02006
.35365
.02258
.40212
.02524
.44796
.02805
12 18
.30314
.02010
.35449
.02262
.40290
.02529
.44870
.02810
Jff 19
.30402
.02014
.35532
.02266
.40369
.02533
.44944
.02815
SO 20
8.30490
0.02018
8.35614
0.02271
8.40447
0.02538
8.45018
0.02820
24 21
.30578
.02022
.35697
.02275
.40525
.02542
.45093
.02824
25 22
.30666
.02026
.35780
.02279
.40603
.02547
.45167
.02829
32 23
.30754
.02030
.35863
.02284
.40681
.02552
.45241
.02834
36 24
.30842
.02034
.35945
.02288
.40760
.02556
.45315
.02839
40 25
8.30929
0.02038
8.36028
0.02292
8.40837
0.02561
8.45388
0.02844
44 26
.31017
.02043
.36110
.02297
.40915
.02565
.45462
.02849
4S 27
.31104
.02047
.36193
.02301
.40993
.02570
.45536
.02853
52 28
.31192
.02051
.36275
.02305
.41071 .
.02575
.45610
.02858
Jtf 29
.31279
.02055
.36357
.02310
.41149
.02579
.45683
.02863
s '
lh Q™ 16°
Ik 10™ 17°
Ik 14™ 18°
Ik 18m 19°
0 30
8.31366
0.02059
8.36439
0.02314
8.41226
0.02584
8.45757
0.02868
4 31
.31453
.02063
.36521
.02319
.41304
.02588
.45830
.02873
8 32
.31540
.02067
.36603
.02323
.41381
.02593
.45904
.02878
i£ 33
.31627
.02071
.36685
.02327
.41459
.02598
.45977
.02883
iff 34
.31714
.02076
.36767
.02332
.41536
.02602
.46050
.02887
20 35
8.31800
0.02080
8.36849
0.02336
8.41613
0.02607
8.46124
0.02892
#4 36
.31887
.02084
.36930
.02340
.41690
.02612
.46197
.02897
28 37
.31974
.02088
.37012
.02345
.41767
.02616
.46270
.02902
32 38
.32060
.02092
.37093
.02349
.41845
.02621
.46343
.02907
Sff 39
.32147
.02096
.37175
.02354
.41921
.02826
.46416
.02912
40 40
8.32233
0.02101
8.37256
0.02358
8.41998
0.02630
8.46489
0.02917
44 41
.32319
.02105
.37337
.02363
.42075
.02635
.46562
. 02922
48 42
.32405
.02109
.37419
.02367
.42152
.02639
.46634
.02926
52 43
.32491
.02113
.37500
.02371
.42229
.02644
.46707
.02931
5<S 44
.32577
.02117
.37581
.02376
.42305
.02649
.'-6780
.02936
s '
Ik 7^ 16°
Ik 11>" 17°
Ik 15m 18°
Ik 19™ 19°
0 45
8.32663
0.02121
8.37662
0.02380
8.42382
0.02653
8.46852
0.02941
4 46
.32749
.02126
.37742
.02385
.42458
.02658
.46925
.02946
S 47
.32834
.02130
.37823
.02389
.42535
.02663
.46998
.02951
J2 48
.32920
.02134
.37904
.02394
.42611
.02668
.47070
.02956
16 49
.33006
.02138
.37985
.02398
.42687
.02672
.47142
.02961
SO 50
8.33091
0.02142
8.38065
0.02402
8.42764
0.02677
8.47215
0.02966
24 51
.33176
.02147
.38146
.02407
.42840
.02682
.47287
.02971
SS 52
.33262
.02151
.38226
.02411
.42916
.02686
.47359
.02976
32 53
.33347
.02155
.38306
.02416
.42992
.02691
.47431
.02981
36 54
.33432
.02159
.38387
.02420
.43068
.02696
.47503
.02986
40 55
8.33517
0.02164
8.38467
0.02425
8.43144
0.02700
8.47575
0.02991
44 56
.33602
.02168
.38547
.02429
.43219
.02705
.47647
.02996
48 57
.33686
.02172
.38627
.02434
.43295
.02710
.47719
.03000
52 58
.33771
.02176
.38707
.02438
.43371
.02715
.47791
.03005
56 59
.33856
.02181
.38787
.02443
.43446
.02719
.47862
.03010
60 60
8.33940
0.02185
8.38867
0.02447
8.43522
0.02724
8.47934
0.03015
Table 10. Haversine Table
255
s '
lh 20m 20°
lh 24™ 21°
lh 28™ 22°
jh Sgm 23°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
8.47934
0.03015
8.52127
0.03321
8.56120
0.03641
8.59931
0.03975
4 1
.48006
.03020
.52195
.03326
.56185
.03646
.59993
.03980
8 2
.48077
.03025
.52263
.03331
.56250
.03652
.60055
.03986
12 3
.48149
.03030
.52331
.03337
.56315
.03657
.60117
.03992
16 4
.48220
.03035
.52399
.03342
.56379
.03663
.60179
.03998
20 5
8.48292
0.03040
8.52467
0.03347
8.56444
0.03668
8.60241
0.04003
24 6
.48363
.03045
.52535
.03352
.56509
.03674
.60303
.04009
28 7
.48434
.03050
.52602
.03358
.56574
.03679
.60365
.04015
32 8
.48505
.03055
.52670
.03363
.56638
.03685
.60426
.04020
36 9
.48576
.03060
.52738
.03368
.56703
.03690
.60488
.04026
40 10
8.48648
0.03065
8.52806
0.03373
8.56767
0.03695
8.60550
0.04032
44 11
.48719
.03070
.52873
.03379
.56832
.03701
.60611
.04038
48 12
.48789
.03075
.52941
.03384
.56896
.03706
.60673
.04043
52 13
.48860
.03080
.53008
.03389
.56960
.03712
.60734
.04049
56 14
.48931
.03085
.53076
.03394
.57025
.03717
.60796
.04055
s '
lh 21m 20°
lh 25m 21°
lh 29™ 22°
jh Sgm 23°
0 15
8.49002
0.03090
8.53143
0.03400
8.57089
0.03723
8.60857
0.04060
4 16
.49073
.03095
.53210
.03405
.57153
.03728
.60919
.04066
S 17
.49143
.03101
.53277
.03410
.57217
.03734
.60980
.04072
12 18
.49214
.03106
.53345
.03415
.57282
.03740
.61041
.04078
iff 19
.49284
.03111
.53412
.03421
.57346
.03745
.61103
.04083
£0 20
8.49355
0.03116
8.53479
0.03426
8.57410
0.03751
8.61164
0.04089
24 21
.49425
.03121
.53546
.03431
.57474
.03756
.61225
.04095
2S 22
.49496
.03126
.53613
.03437
.57538
.03762
.61286
.04101
32 23
.49566
.03131
.53680
.03442
.57601
.03767
.61347
.04106
30 24
.49636
.03136
.53747
.03447
.57665
.03773
.61408
.04112
40 25
8.49706
0.03141
8.53814
0.03453
8.57729
0.03778
8.61469
0.04118
44 26
.49777
.03146
.53880
.03458
.57793
.03784
.61530
.04124
45 27
.49847
.03151
.53947
.03463
.57856
.03789
.61591
.04130
52 28
.49917
.03156
.54014
.03468
.57920
.03795
.61652
.04135
56 29
.49987
.03161
.54080
.03474
.57984
.03800
.61713
.04141
s '
lh 22m 20°
lh gffi* 21°
lh som 22°
/* 34m 23°
0 30
8.50056
0.03166
8.54147
0.03479
8.58047
0.03806
8.61773
0.04147
^ 31
.50126
.03171
.54214
.03484
.58111
.03812
.61834
.04153
8 32
.50196
.03177
.54280
.03490
.58174
.03817
.61895
.04159
10 33
.50266
.03182
.54346
.03495
.58238
.03823
.61955
.04164
/ff 34
.50335
.03187
.54413
.03500
.58301
.03828
.62016
.04170
20 35
8.50405
0.03192
8.54479
0.03506
8.58364
0.03834
8.62077
0.04176
24 36
.50475
.03197
.54545
.03511
.58427
.03839
.62137
.04182
28 37
.50544
.03202
.54612
.03517
.58491
.03845
.62197
.04188
32 38
.50614
.03207
.54678
.03522
.58554
.03851
.62258
.04194
36 39
.50683
.03212
.54744
.03527
.58617
.03856
.62318
.04199
40 40
8.50752
0.03218
8.54810
0.03533
8.58680'
0.03862
8.62379
0.04205
44 41
.50821
.03223
.54876
.03538
.58743
.03867
.62439
.04211
48 42
.50891
.03228
.54942
.03543
.58806
.03873
.62499
.04217
52 43
.50960
.03233
.55008
.03549
.58869
.03879
.62559
.04223
5ff 44
.51029
.03238
.55073
.03554
.58932
.03884
.62619
.04229
s '
lh 23™ 20°
lh 27m 21°
lh Sim 22°
lh 35m 23°
0 45
8.51098
0.03243
8.55139
0.03560
8.58994
0.03890
8.62680
0.04234
4 46
.51167
.03248
.55205
.03565
.59057
.03896
.62740
.04240
S 47
.51236
.03254
.55271
.03570
.59120
.03901
.62800
.04246
J2 48
.51305
.03259
.55336
.03576
.59183
.03907
.62860
.04252
16 49
.51374
.03264
.55402
.03581
.59245
.03912
.62919
.04258
20 50
8.51442
0.03269
8.55467
0.03587
8.59308
0.03918
8.62979
0.04264
24 51
.51511
.03274
.55533
.03592
.59370
.03924
.63039
.04270
25 52
.51580
.03279
.55598
.03597
.59433
.03929
.63099
.04276
32 53
.51648
.03285
.55664
.03603
.59495
.03935
.63159
.04281
Sff 54
.51717
.03290
.55729
.03608
.59558
.03941
.63218
.04287
40 55
8.51785
0.03295
8.55794
0.03614
8.59620
0.03946
8.63278
0.04293
44 56
.51854
.03300
.55859
.03619
.59682
.03952
.63338
.04299
4S 57
.51922
.03305
.55925
.03624
.59745
.03958
.63397
.04305
.52 58
.51990
.03311
.55990
.03630
.59807
.03963
.63457
.04311
56 59
.52058
.03316
.56055
.03635
.59869
.03969
.63516
.04317
60 60
8.52127
0.03321
8.56120
0.03641
8.59931
0.03975
8.63576
0.04323
256
Table 10. Haversine Table
s '
lh Sffn 24°
1>> 40m 25°
I* 44m 26°
lh .££>» 27°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
8.63576
0.04323
8.67067
0.04685
8.70418
0.05060
8.73637
0.05450
4 1
.63635
.04329
.67124
.04691
.70472
.05067
.73690
.05456
8 2
.63695
.04335
.67181
.04697
.70527
.05073
.73742
.05463
12 3
.63754
.04340
.67238
.04703
.70582
.05079
.73795
.05470
16 4
.63813
.04346
.67295
.04709
.70636
.05086
.73847
.05476
20 5
8.63872
0.04352
8.67352
0.04715
8.70691"
0.05092
8.73900
0.05483
24 6
.63932
.04358
.67409
.04722
.70745
.05099
.73952
.05489
28 7
-.63991
.04364
.67465
.04728
.70800
.05105
.74005
.05496
32 8
.64050
.04370
.67522
.04734
.70854
.05111
.74057
.05503
36 9
.64109
.04376
.67579
.04740
.70909
.05118
.74109
.05509
40 10
8.64168
0.04382
8.67635
0.04746
8.70963
0.05124
8.74162
0.05516
44 11
.64227
.04388
.67692
.04752
.71017
.05131
.74214
.05523
48 12
.64286
.04394
.67748
.04759
.71072
.05137
.74266
.05529
52 13
.64345
.04400
.67805
.04765
.71126
.05144
.74318
.05536
56 14
.64404
.04405
.67861
.04771
.71180
.05150
.74371
.05542
s '
lh 3jm 24°
lh um 25o
1* 45m 26°
lh ^y» 27°
0 15
8.64463
0.04412
8.67918
0.04777
8.71234
0.05156
8.74423
0.05549
', 16
.64521
.04418
.67974
.04783
.71289
.05163
.74475
.05556
S 17
.64580
.04424
.68030
.04790
.71343
.05169
.74527
.05562
12 18
.64639
.04430
.68087
.04796
.71397
.05176
.74579
.05569
J6 19
.64697
.04436
.68143
.04802
.71451
.05182
.74631
.05576
20 20
8.64756
0.04442
8.68199
0.04808
8.71505
0.05189
8.74683
0.05582
24 21
.64815
.04448
.68256
.04815
.71559
.05195
.74735
.05589
2S 22
.64873
.04454
.68312
.04821
.71613
.05201
.74787
.05596
32 23
.64932
.04460
.68368
.04827
.71667
.05208
.74839
.05603
36 24
.64990
.04466
.68424
.04833
.71721
.05214
.74890
.05609
40 25
8.65049
0.04472
8.68480
0.04839
8.71774
0.05221
8.74942
0.05616
44 26
.65107
.04478
.68536
.04846
.71828
.05227
.74994
.05623
48 27
.65165
.04484
.68592
.04852
.71882
.05234
.75046
.05629
52 28
.65224
.04490
.68648
.04858
.71936
.05240
.75097
.05636
56 29
.65282
.04496
.68704
.04864
.71989
.05247
.75149
.05643
s '
lh 28m 24°
lh .££>« 25°
1* 4&» 26°
1* 50™ 27°
0 30
8.65340
0.04502
8.68760
0.04871
8.72043
0.05253
8.75201
0.05649
4 31
.65398
.04508
.68815
.04877
.72097
.05260
.75252
.05656
8 32
.65456
.04514
.68871
.04883
.72150
.05266
.75304
.05663
.72 33
.65514
.04520
.68927
.04890
.72204
.05273
.75355
.05670
^6 34
.65572
.04526
.68983
.048%
.72257
.05279
.75407
.05676
20 35
8.65630
0.04532
8.69038
0.04902
8.72311
0.05286
8.75458'
0.05683
24 36
.65688
.04538
.69094
.04908
.72364
.05292
.75510
.05690
28 37
.65746
.04544
.69149
.04915
.72418
.05299
.75561
.05697
32 38
.65804
.04550
.69205
.04921
.72471
.05305
.75613
.05703
36 39
.65862
.04556
.69260
.04927
.72525
.05312
.75664
.05710
40 40
8.65920
0.04562
8.69316
0.04934
8.72578
0.05318
8.75715
0.05717
44 41
.65978
.04569
.69371
.04940
.72631
.05325
.75767
.05724
48 42
.66035
.04575
.69427
.04946
.72684
.05331
.75818
.05730
52 43
.66093
.04581
.69482
.04952
.72738
.05338
.75869
.05737
56 44
.66151
.04587
.69537
.04959
.72791
.05345
:, 5920
.05744
s '
lh gym 24°
lh 43™ 25°
Jh 47m 26°
lh 51*. 27°
0 45
8.66208
0.04593
8.69593
0.04965
8.72844
0.05351
8.75972
O.C5751
4 46
.66266
.04599
.69648
.04971
.72897
.05358
.76023
.05757
S 47
.66323
.04605
.69703
.04978
.72950
.05364
.76074
.C5764
.72 48
.66381
.04611
.69758
.04984
.73003
.05371
.76125
.05771
16 49
.66438
.04617
.69814
.04990
.73056
.05377
.76176
.05778
20 50
8.66496
0.04623
8.69869
0.04997
8.73109
0.05384
8.76227
O.C5785
24 51
.66553
.04629
.69924
.05003
.73162
.05390
.76278
.05791
28 52
.66610
.04636
.69979
.05009
.73215
.05397
.76329
.05798
32 53
.66668
.04642
.70034
.05016
.73268
.05404
.76380
.05805
36 54
.66725
.04648
.70089
.05022
.73321
.05410
.76431
.05812
40 55
8.66782
0.04654
8.70144
0.05028
8.73374
0.05417
8.76481
0.05819
44 56
.66839
.04660
.70198
.05035
.73426
.05423
.76532
.05825
48 57
.66896
.04666
.70253
.05041
.73479
.05430
.76583
.05832
52 58
.66953
.04672
.70308
.05048
.73532
.05436
.76634
.05839
56 59
.67010
.04678
.70363
.05054
.73584
.05443
.76684
.05846
60 60
8.67067
0.04685
8.70418
0.05060
8.73637
0.05450
8.76735
0.05853
Table 10. Haversine Table
257
s '
7* 52m 28°
J* 56'" 29°
2h om 30°
2* 4m 31°
Hav.
No.
Hav. No.
Hav.
No.
Hav.
No.
0 0
8.76735
0.05853
8.79720
0.06269
8.82599
0.06699
8.85380
0.07142
4 1
.76786
.05859
.79769
.06276
.82646
.06706
.85425
.07149
8 2
.76836
.05866
.79818
.06283
.82694
.06713
.85471
.07157
12 3
.76887
.05873
.79866
.06290
.82741
.06721
.85516
.07164
16 4
.76938
.05880
.79915
.06297
.82788
.06728
.85562
.07172
20 5
8.76988
0.05887
8.79964
0.06304
8.82835
0.06735
8.85607
0.07179
24 6
.77039
.05894
.80013
.06311
.82882
.06742
.85653
.07187
28 7
.77089
.05901
.80061
.06318
.82929
.06750
.85698
.07194
32 8
.77139
.05907
.80110
.06326
.82976
.06757
.85743
.07202
36 9
.77190
.05914
.80158
.06333
.83023
.06764
.85789
.07209
40 10
8.77240
0.05921
8.80207
0.06340
8.83069
0.06772
8.85834
0.07217
44 11
.77291
.05928
.80256
.06347
.83116
.06779
.85879
.07224
48 12
.77341
.05935
.80304
.06354
.83163
.06786
.85925
.07232
52 13
.77391
.05942
.80353
.06361
.83210
.06794
.85970
.07239
56 14
.77441
.05949
.80401
.06368
.83257
.06801
.86015
.07247
s '
7* 53m 28°
lh j7»> 29°
2* lm 30°
2h 5m 31°
0 15
8.77492
0.05955
8.80449
0.06375
8.83303
0.06808
8.86060
0.07254
4 16
.77542
.05982
.80498
.06382
.83350
.06816
.86105
.07262
S 17
.77592
.05969
.80546
.06389
.83397
.06823
.86151
.07270
12 18
.77642
.05976
.80595
.06397
.83444
.06830
.86196
.07277
J6 19
.77692
.05983
.80643
.06404
.83490
.06838
.86241
.07285
20 20
8.77742
0.05990
8.80691
0.06411
8.83537
0.06845
8.86286
0.07292
24 21
.77792
.05997
.80739
.06418
.83583
.06852
.86331
.07300
2S 22
.77842
.06004
.80788
.06425
.83630
.06860
.86376
.07307
32 23
.77892
.06011
.80836
.06432
.83676
.06867
.86421
.07315
36 24
.77942
.06018
.80884
.06439
.83723
.06874
.86466
.07322
40 25
8.77992
0.06024
8.80932
0.06446
8.83769
0.06882
8.86511
0.07330
44 26
.78042
.06031
.80980
.06454
.83816
.06889
.86556
.07338
45 27
.78092
.06038
.81028
.06461
.83862
.06896
.86600
.07345
52 28
.78142
.06045
.81076
.06468
.83909
.06904
.86645
.07353
56 29
.78191
.06052
.81124
.06475
.83955
.06911
.86690
.07360
s '
1* 54m 28°
lh o8m 29°
gh gm 3Q°
2h Qm 31°
0 30
8.78241
0.06059
8.81172
0.06482
8.84002
0.06919
8.86735
0.07368
4 31
.78291
.06066
.81220
.06489
.84048
.06926
.86780
.07376
8 32
.78341
.06073
.81268
.06497
.84094
.06933
.86825
.07383
72 33
.78390
.06080
.81316
.06504
.84140
.06941
.86869
.07391
16 34
.78440
.06087
.81364
.06511
.84187
.06948
.86914
.07398
20 35
8.78490
0.06094
8.81412
0.06518
8.84233
0.06956
8.86959
0.07406
24 36
.78539
.06101
.81460
.06525
.84279
.06963
.87003
.07414
28 37
.78589
.06108
.81508
.06532
.84325
.06970
.87048
.07421
32 38
.78638
.06115
.81555
.06540
.84371
.06978
.87093
.07429
36 39
.78688
.06122
.81603
.06547
.84417
.06985
.87137
.07437
40 40
8.78737
0.06129
8.81651
0.06554
8.84464
0.06993
8.87182
0.07444
44 41
.78787
.06136
.81699
.06561
.84510
.07000
.87226
.07452
48 42
.78836
.06143
.81746
.06568
.84556
.07007
.87271
.07459
52 43
.78885
.06150
.81794
.06576
.84602
.07015
.87315
.07467
56 44
.78935
.06157
.81841
.06583
.84648
07022
.87360
.07475
s . '
Ik 55m 28°
lh 59* 29°
2h 3"1 30°
2h 7« 31°
0 45
8.78984
0.06164
8.81889
0.06590
8.84694
0.07030
8.87404
0.07482
4 46
.79033
.06171
.81937
.06597
.84740
.07037
.87448
.07490
S 47
.79082
.06178
.81984
.06605
.84785
.07045
.87493
.07498
12 48
.79132
.06185
.82032
.06612
.84831
.07052
.87537
.07505
16 49
.79181
.06192
.82079
.06619
.84877
.07059
.87582
.07513
20 50
8.79230
0.06199
8.82126
0.06626
8.84923
0.07067
8.87626
0.07521
24 51
.79279
.06206
.82174
.06633
.84969
.07074
.87670
.07528
28 52
.79328
.06213
.82221
.06641
.85015
.07082
.87714
.07536
32 63
.79377
.06220
.82269 .
.06648
.85060
.07089
.87759
.07544
36 54
.79426
.06227
.82316
.06655
.85106
.07097
.87803
.07551
40 65
8.79475
0.06234
8.82363
0.06662
8.85152
0.07104
8.87847
0.07559
44 56
.79524
.06241
.82410
.06670
.85197
.07112
.87891
.07567
48 57
.79573
.06248
.82458
.06677
.85243
.07119
.87935
.07574
5J 58
.79622
.06255
.82505
.06684
.85289
.07127
.87980
.07582
56 59
.79671
.06262
.82552
.06691
.85334
.07134
.88024
.07590
60 60
8.79720
0.06269
8.82599
0.06699
8.85380
0.07142
8.88068
0.07598
258
Table 10. Haversine Table
s '
%h g™ 32°
2h 12™ 33°
2h 16m 34°
2* 20m 35°
HOT.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
8.88068
0.07598
8.90668
0.08066
8.93187
0.08548
8.95628
0.09042
4 1
.88112
.07605
.90711
.08074
.93228
.08556
.95668
.09051
8 2
.88156
.07613
.90754
.08082
.93270
.08564
.95709
.09059
12 3
.88200
.07621
.90796
.08090
.93311
.08573
.95749
.09067
18 4
.88244
.07628
.90839
.08098
.93352
.08581
.95789
.09076
20 5
8.88288
0.07636
8.90881
0.08106
8.93393
0.08589
8.95828
0.09084
24 6
.88332
.07644
.90924
.08114
.93435
.08597
.95868
.09093
28 7
.88375
.07652
.90966
.08122
.93476
.08605
.95908
.09101
32 8
.88419
.07659
.91009
.08130
.93517
.08613
.95948
.09109
36 9
.88463
.07667
.91051
.08138
.93558
.08621
.95988
.09118
40 10
8.88507
0.07675
8.91094
0.08146
8.93599
0.08630
8.96028
0.09126
44 11
.88551
.07683
.91136
.08154
.93640
.08638
.96068
.09134
48 12
.88595
.07690
.91179
.08162
.93681
.08646
.96108
.09143
52 13
.88638
.07698
.91221
.08170
.93722
.08654
.96148
.09151
56 14
.88682
.07706
.91263
.08178
.93764
.08662
.96187
.09160
s '
2h Sf"1 32°
2h IS"1 33°
2h 17m 34°
%h 21m 35°
0 15
8.88726
0.07714
8.91306
0.08186
8.93805
0.08671
8.96227
0.09168
4 16
.88769
.07721
.91348
.08194
.93846
.08679
.96267
.09176
5 17
.88813
.07729
.91390
.08202
.93886
.08687
.96307
.09185
12 18
.88857
.07737
.91432
.08210
.93927
.08695
.96346
.09193
iff 19
.88900
.07745
.91475
.08218
.93968
.08703
.96386
.09202
£0 20
8.88944
0.07752
8.91517
0.08226
8.94009
0.08711
8.96426
0.09210
24 21
.88988
.07760
.91559
.08234
.94050
.08720
.96465
.09218
25 22
.89031
.07768
.91601
.08242
.94091
.08728
.96505
.09227
32 23
.89075
.07776
.91643
.08250
.94132
.08736
.96545
.09235
35 24
.89118
.07784
.91685
.08258
.94173
.08744
.96584
.09244
40 25
8.89162
0.07791
8.91728
0.08266
8.94213
0.08753
8.96624
0.09252
44 26
.89205
.07799
.91770
.08274
.94254
.08761
.96663
.09260
45 27
.89248
.07807
.91812
.08282
.94295
.08769
.96703
.09269
52 28
.89292
.07815
.91854
.08290
.94336
.08777
.96742
.09277
56 29
.89335
.07823
.91896
.08298
.94376
.08785
.96782
.09286
s '
gh wm 32°
2h 14m 33°
2?A 18m 34°
%h 22"1 35°
0 30
8.89379
0.07830
8.91938
0.08306
8.94417
0.08794
8.96821
0.09294
4 31
.89422
.07838
.91980
.08314
.94458
.08802
.96861
.09303
8 32
.89465
.07846
.92022
.08322
.94498
.08810
.96900
.09311
12 33
.89509
.07854
.92064
.08330
.94539
.08818
.96940
.09320
15 34
.89552
.07862
.92105
.08338
.94580
.08827
.96979
.09328
20 35
8.89595
0.07870
8.92147
0.08346
8.94620
0.08835
8.97018
0.09337
24 36
.89638
.07877
.92189
.08354
.94661
.08843
.97058
.09345
28 37
.89681
.07885
.92231
.08362
.94701
.08851
.97097
.09353
32 38
.89725
.07893
.92273
.08370
.94742
.08860
.97136
.09362
35 39
.89768
.07901
.92315
.08378
.94782
.08868
.97176
.09370
40 40
8.89811
0.07909
8.92356
0.08386
8.94823
0.08876
8.97215
0.09379
44 41
.89854
.07917
.92398
.08394
.94863
.08885
.97254
.09387
48 42
.89897
.07924
.92440
.08402
.94904
.08893
.97294
.09396
52 43
.89940
.07932
.92482
.08410
.94944
.08901
.97333
.09404
55 44
.89983
.07940
.92523
.08418
.94985
.08909
97372
.09413
s '
2h llm 32°
2k 15m 33°
2h 19m 34°
2h 23™ 35°
0 45
8.90026
0.07948
8.92565
0.08427
8.95025
0.08918
8.97411
0.09421
4 46
.90069
.07956
.92607
.08435
.95065
.08926
.97450
.09430
S 47
.90112
.07964
.92648
.08443
.95106
.08934
.97489
.09438
12 48
.90155
.07972
.92690
.08451
.95146
.08943
.97529
.09447
16 49
.90198
.07980
.92731
.08459
.95186
.08951
.97568
.09455
20 50
8.90241
0.07987
8.92773
0.08467
8.95227
0.08959
8.97607
0.09464
24 51
.90284
.07995
.92814
.08475
.95267
.08967
.97646
.09472
28 52
.90326
.08003
.92856
.08483
.95307
.08976
.97685
.09481
32 53
.90369
.08011
.92897
.08491
.95347
.08984
.97724
.09489
35 54
.90412
.08019
.92939
.08499
.95388
.08992
.97763
.09498
40 55
8.90455
0.08027
8.92980
0.08508
8.95428
0.09001
8.97802
0.09506
44 56
.90498
.08035
.93022
.08516
.95468
.09009
.97841
.09515
48 57
.90540
.08043
.93063
.08524
.95508
.09017
.97880
.09524
52 58
.90583
.08051
.93104
.08532
.95548
.09026
.97919
.09532
55 59
.90626
.08059
.93146
.08540
.95588
.09034
.97958
.09541
60 60
8.90668
0.08066
8.93187
0.08548
8.95628
0.09042
8.97997
0.09549
Table 10. Harersine Table
259
s '
2* 24m 36°
2h 28™ 37°
2h 32m 38°
2h 36m 39°
Bar.
No.
Bav.
No.
Bar.
No.
Bav.
No.
0 0
8.97997
0.09549
9.00295
0.10068
9.02528
0.10599
9.04699
0.11143
4 1
.98035
.09558
.00333
.10077
.02565
.10608
.04735
.11152
8 2
.98074
.09566
.00371
.10086
.02602
.10617
.04770
.11161
12 3
.98113
.09575
.00408
.10095
.02638
.10626
.04806
.11170
16 4
.98152
.09583
.00446
.10103
.02675
.10635
.04842
.11179
20 5
8.98191
0.09592
9.00484
0.10112
9.02712
0.10644
9.04877
0.11189
24 6
.98229
.09601
.00522
.10121
.02748
.10653
.04913
.11198
28 7
.98268
.09609
.00559
.10130
.02785
.10662
.04948
.11207
32 8
.98307
.09618
.00597
.10138
.02821
.10671
.04984
.11216
36 9
.98346
.09626
.00634
.10147
.02858
.10680
.05019
.11225
40 10
8.98384
0.09635
9.00672
0.10156
9.02894
0.10689
9.05055
0.11234
44 11
.98423
.09643
.00710
.10165
.02931
.10698
.05090
.11244
48 12
.98462
.09652
.00747
.10174
.02967
.10707
.05126
.11253
52 13
.98500
.09661
.00785
.10182
.03004
.10716
.05161
.11262
56 14
.98539
.09669
.00822
.10191
.03040
.10725
.05197
.11271
s '
2* 25m 36°
2h 29m 37°
2h 33m 38°
2* 37m 39°
0 15
8.98578
0.09678
9.00860
0.10200
9.03077
0.10734
9.05232
0.11280
4 16
.98616
.09686
.00897
.10209
.03113
.10743
.05268
.11290
S 17
.98655
.09695
.00935
.10218
.03150
.10752
.05303
.11299
12 18
.98693
.09704
.00972
.10226
.03186
.10761
.05339
.11308
7<5 19
.98732
.09712
.01009
.10235
.03222
.10770
.05374
.11317
£0 20
8.98770
0.09721
9.01047
0.10244
9.03259
0.10779
9.05409
0.11326
24 21
.98809
.09729
.0-1084
.10253
.03295
.10788
.05445
.11336
25 22
.'.ISM7
.09738
.01122
.10262
.03331
.10797
.05480
.11345
32 23
.DSSS6
.09747
.01159
.10270
.03368
.10806
.05515
.11354
36 24
.98924
.09755
.01196
.10279
.03404
.10815
.05551
.11363
40 25
8.98963
0.09764
9.01234
0.10288
9.03440
0.10824
9.05586
0.11373
44 26
.99001
.09773
.01271
.10297
.03476
.10833
.05621
.11382
4S 27
.99039
.09781
.01308
.10306
.03513
.10842
.05656
.11391
52 28
.99078
.09790
.01345
.10315
.03549
.10851
.05692
.11400
56 29
.99116
.09799
.01383
.10323
.03585
.10861
.05727
.11410
s '
2* 261" 36°
2* SO™ 37°
2h 34m 38°
2* 38™ 39°
0 30
8.99154
0.09807
9.01420
0.10332
9.03621
0.10870
9.05762
0.11419
4 31
.99193
.09816
.01457
.10341
.03657
.10879
.05797
.11428
8 32
.99231
.09824
.01494
.10350
.03694
.10888
.05832
.11437
.72 33
.99269
.09833
.01531
.10359
.03730
.10897
.05867
.11447
16 34
.99307
.09842
.01569
.10368
.03766
.10906
.05903
.11456
20 35
8.99346
0.09850
9.01606
0.10377
9.03802
0.10915
9.05938
0.11465
24 36
.99384
.09859
.01643
.10386
.03838
.10924
.05973
.11474
28 37
.99422
.09868
.01680
.10394
.03874
.10933
.06008
.11484
32 38
.99460
.09876
.01717
.10403
.03910
.10942
.06043
.11493
36 39
.99498
.09885
.01754
.10412
.03946
.10951
.06078
.11502
40 40
8.99536
0.09894
9.01791
0.10421
9.03982
0.10960
9.06113
0.11511
44 41
.99575
.09903
.01828
.10430
.04018
.10969
.06148
.11521
48 42
.99613
.09911
.01865
.10439
.04054
.10978
.06183
.11530
52 43
.99651
.09920
.01902
.10448
.04090
.10988
.06218
.11539
56 44
.99689
.09929
.01939
.10457
.04126
.10997
.0(1253
.11549
8 '
2* 27m 36°
2h 31m 37°
2* 35m 38°
2* 39m 39°
0 45
8.99727
0.09937
9.01976
0.10466
9.04162
0.11006
9.06288
0.11558
4 46
.99765
.09946
.02013
.10474
.04198
.11015
.06323
.11567
S 47
.99803
.09955
.02050
.10483
.04234
.11024
.06358
.11577
.72 48
.99841
.09963
.02087
.10492
.04270
.11033
.06393
.11586
16 49
.99879
.09972
.02124
.10501
.04306
.11042
.06428
.11595
20 50
8.99917
0.09981
9.02161
0.10510
9.04341
0.11051
9.06462
0.11604
24 51
.99955
.09990
.02197
.10519
.04377
.11060
.06497
.11614
28 52
.99993
.09998
.02234
.10528
.04413
.11070
.06532
.11623
32 53
9.00031
.10007
.02271
.10637
.04449
.11079
.06567
.11632
36 54
.00068
.10016
.02308
.10546
.04485
.11088
.06602
.11642
40 55
9.00106
0.10025
9.02345
0.10555
9.04520
0.11097
9.06637
0.11651
44 56
.00144
.10033
.02381
.10564
.04556
.11106
.06871
.11660
48 57
.00182
.10042
.02418
.10573
.04592
.11115
.06706
.11670
52 58
.00220
.10051
.02455
.10582
.04628
.11124
.06741
.11679
56 59
.00258
.10059
.02492
.10591
.04663
.11134
.06776
.11688
60 60
9.00295
010068
9.02528
0.10599
9.04699
0.11143
9.06810
0.11698
260
Table 10. Haversine Table
s '
2h 40m 40°
2* 44m 41°
2h 48™ 42°
2* 52" 43°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.06810
0.11698
9.08865
0.12265
9.10866
0.12843
9.12815
0.13432
4 1
.06845
.11707
.08899
.12274
.10899
.12852
.12847
.13442
5 2
.06880
.11716
.08933
.12284
.10932
.12862
.12879
.13452
12 3
.06914
.11726
.08966
.12293
.10965
.12872
.12911
.13462
16 4
.06949
.11735
.09000
.12303
.10997
.12882
.12943
.13472
20 5
9.06984
0.11745
9.09034
0.12312
9.11030
0.12891
9.12975
0.13482
24 6
.07018
.11754
.09068
.12322
.11063
.12901
.13007
.13492
28 7
.07053
.11763
.09101
.12331
.11096
.12911
.13039
.13502
32 8
.07088
.11773
.09135
.12341
.11129
.12921
.13071
.13512
'36 9
.07122
.11782
.09169
.12351
.11161
.12930
.13103
.13522
40 10
9.07157
0.11791
9.09202
0.12360
9.11194
0.12940
9.13135
0.13532
44 11
.07191
.11801
.09236
.12370
.11227
.12950
.13167
.13542
48 12
.07226
.11810
.09269
.12379
.11260
.12960
.13199
.13552
52 13
.07260
.11820
.09303
.12389
.11292
.12970
.13231
.13562
56 14
.07295
.11829
.09337
.12398
.11325
.12979
.13263
.13571
s '
2h 41m 40°
2* 45TO 41°
2h 49m 42°
2* 53"1 43°
0 15
9.07329
0.11838
9.09370
0.12408
9.11358
0.12989
9.13295
0.13581
4 16
.07364
.11848
.09404
.12418
.11391
.12999
.13326
.13591
S 17
.07398
.11857
.09437
.12427
.11423
.13009
.13358
.13601
12 18
.07433
.11867
.09471
.12437
.11456
.13018
.13390
.13611
.70 19
.07467
.11876
.09504
.12446
.11489
.13028
.13422
.13621
20 20
9.07501
0.11885
9.09538
0.12456
9.11521
0.13038
9.13454
0.13631
24 21
.07536
.11895
.09571
.12466
.11554
.13048
.13486
.13641
2S 22
.07570
.11904
.09605
.12475
.11586
.13058
.13517
.13651
32 23
.07605
.11914
.09638
.12485
.11619
.13067
.13549
.13661
36 24
.07639
.11923
.09672
.12494
.11652
.13077
.13581
.13671
40 25
9.07673
0.11933
9.09705
0.12504
9.11684
0.13087
9.13613
0.13681
44 26
.07708
.11942
.09739
.12514
.11717
.13097
.13644
.13691
48 27
.07742
.11951
.09772
.12523
.11749
.13107
.13676
.13701
52 28
.07776
.11961
.09805
.12533
.11782
.13116
.13708
.13711
56 29
.07810
.11970
.09839
.12543
.11814
.13126
.13739
.13721
s '
2h 42™ 40°
2h 4&m 41°
2* 50m 42°
2* 54m 43°
0 30
9.07845
0.11980
9.09872
0.12552
9.11847
0.13136
9.13771
0.13731
4 31
.07879
.11989
.09905
.12562
.11879
.13146
.13803
.13741
8 32
.07913
.11999
.09939
.12572
.11912
.13156
.13834
.13751
^2 33
.07947
.12008
.09972
.12581
.11944
.13166
.13866
.13761
/0 34
.07981
.12018
.10005
.12591
.11977
.13175
.13898
.13771
20 35
9.08016
0.12027
9.10039
0.12600
9.12009
0.13185
9.13929
0.13781
24 36
.08050
.12036
.10072
.12610
.12041
.13195
.13961
.13791
28 37
.08084
.12046
.10105
.12620
.12074
.13205
.13992
.13801
32 38
.08118
.12055
.10138
.12629
.12106
.13215
.14024
.13811
30 39
.08152
.12065
.10172
.12639
.12139
.13225
.14056
.13822
40 40
9.08186
0.12074
9.10205
0.12649
9.12171
0.13235
9.14087
0.13832
44 41
.08220
.12084
.10238
.12658
.12203
.13244
.14119
.13842
48 42
.08254
.12093
.10271
.12668
.12236
.13254
.14150
.13852
52 43
.08288
.12103
.10304
.12678
.12268
.13264
.14182
.13862
56 44
.08323
.12112
.10337
.12687
.12300
.13274
.44213
.13872
s '
2h 43m 40°
2h J^m. 41°
2h 51m 42°
2h 55m 43°
'/ 45
9.08357
0.12122
9.10371
0.12697
9.12332
0.13284
9.14245
0.13882
4 46
.08391
.12131
.10404
.12707
.12365
.13294
.14276
.13892
S 47
.08425
.12141
.10437
.12717
.12397
.13304
.14307
.13902
.72 48
.08459
.12150
.10470
.12726
.12429
.13314
.14339
.13912
16 49
.08492
.12160
.10503
.12736
.12461
.13323
.14370
.13922
20 50
9.08526
0.12169
9.10536
0.12746
9.12494
0.13333
9.14402
0.13932
24 51
.08560
.12179
.10569
.12755
.12526
.13343
.14433
.13942
28 52
.08594
.12188
.10602
.12765
.12558
.13353
.14465
.13952
32 53
.08628
.12198
.10635
.12775
.12590
.13363
.14496
.13962
36 54
.08662
.12207
.10668
.12784
.12622
.13373
.14527
.13972
40 55
9.08696
0.12217
9.10701
0.12794
9.12655
0.13383
9.14559
0.13983
44 56
.08730
.12226
.10734
.12804
.12687
.13393
.14590
.13993
48 57
.08764
.12236
.10767
.12814
.12719
.13403
.14621
.14003
52 58
.08797
.12245
.10800
.12823
.12751
.13412
.14653
.14013
56 59
.08831
.12255
.10833
.12833
.12783
.13422
.14684
.14023
60 60
9.08865
0 12265
9.10866
0.12843
9.12815
0.13432
9.14715
0.14033
Table 10. Haversine Table
261
s '
2* 5fim 44°
3* Om 45°
SA 4m 46°
3h 8m. 470
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.14715
0.14033
9.16568
0.14645
9.18376
0.15267
9.20140
0.15900
4 1
.14746
.14043
.16598
.14655
.18405
.15278
.20169
.15911
8 2
.14778
.14053
.16629
.14665
.18435
.15288
.20198
.15921
12 3
.14809
.14063
.16659
.14676
.18465
.15298
.20227
.15932
16 4
.14840
.14073
.16690
.14686
.18495
.15309
.20256
.15943
20 5
9.14871
0.14084
9.16720
0.14696
9.18524
0.15319
9.20285
0.15953
24 6
.14902
.14094
.16751
.14706
.18554
.15330
.20314
.15964
28 7
.14934
.14104
.16781
.14717
.18584
.15340
.20343
.15975
32 8
.14965
.14114
.16812
.14727
.18613
.15351
.20372
.15985
36 9
.14996
.14124
.16842
.14737
.18643
.15361
.20401
.15996
40 10
9.15027
0.14134
9.16872
0.14748
9.18673
0.15372
9.20430
0.16007
44 11
.15058
.14144
.16903
.14758
.18702
.15382
.20459
.16017
48 12
.15089
.14154
.16933
.14768
.18732
.15393
.20488
.16028
52 13
.15120
.14165
.16963
.14779
.18762
.15403
.20517
.16039
56 14
.15152
.14175
.16994
.14789
.18791
.15414
.20546
.16049
s '
2h a7m 44°
3h jm 45°
3k sm 46°
Sh 9m 470
0 15
9.15183
0.14185
9.17024
0.14799
9.18821
0.15424
9.20574
0.16060
4 16
.15214
.14195
.17054
.14810
.18850
.15435
.20603
.16071
5 17
.15245
.14205
.17085
.14820
.18880
.15445
.20632
.16081
12 18
.15276
.14215
.17115
.14830
.18909
.15456
.20661
.16092
J6 19
.15307
.14226
.17145
.14841
.18939
.15466
.20690
.16103
20 20
9.15338
0.14236
9.17175
0.14851
9.18968
0.15477
9.20719
0.16113
24 21
.15369
.14246
.17206
.14861
.18998
.15487
.20748
.16124
25 22
.15400
.14256
.17236
.14872
.19027
.15498
.20776
.16135
32 23
.15431
.14266
.17266
.14882
.19057
.15509
.20805
.16146
36 24
.15462
.14276
.17296
.14892
.19086
.15519
.20834
.16156
40 25
9.15493
0.14287
9.17327
0.14903
9.19116
0.15530
9.2Q863
0.16167
44 26
.15524
.14297
.17357
.14913
.19145
.15540
.20891
.16178
4S 27
.15555
.14307
.17387
.14923
.19175
.15551
.20920
.16188
52 28
.15585
.14317
.17417
.14934
.19204
.15561
.20949
.16199
56 29
.15616
.14327
.17447
.14944
.19234
.15572
.20978
.16210
s '
2* 58>n 44°
3A 2"* 45°
3h ffn 45°
3h jo™ 47°
0 30
9.15647
0.14337
9.17477
0.14955
9.19263
0.15582
9.21006
0.16220
4 31
.15678
.14348
.17507
.14965
.19292
.15593
.21035
.16231
8 32
.15709
.14358
.17538
.14975
.19322
.15603
.21064
.16242
12 33
.15740
.14368
.17568
.14986
.19351
.15614
.21092
.16253
/6 34
.15771
.14378
.17598
.14996
.19381
.15625
.21121
.16263
20 35
9.15802
0.14388
9.17628
0.15006
9.19410
0.15635
9.21150
0.16274
24 36
.15832
.14399
.17658
.15017
.19439
.15646
.21178
.16285
28 37
.15863
.14409
.17688
.15027
.19469
.15656
.21207
.16296
32 38
.15894
.14419
.17718
.15038
.19498
.15667
.21236
.16306
36 39
.15925
.14429
.17748
.15048
.19527
.15677
.21264
.16317
40 40
9.15955
0.14440
9.17778
0.15058
9.19557
0.15688
9.21293
0.16328
44 41
.15986
.14450
.17808
.15069
.19586
.15699
.21322
.16339
48 42
.16017
.14460
.17838
.15079
.19615
.15709
.21350
.16349
52 43
.16048
.14470
.17868
.15090
.19644
.15720
.21379
.16360
56 44
.16078
.14480
.17898
.15100
.19674
.15730
.21407
.16371
s '
2A5£m 44°
3h S"1 45°
3* 7« 46°
3h llm 47°
0 45
9.16109
0.14491
9.17928
0.15110
9.19703
0.15741
9.21436
0.16382
4 46
.16140
.14501
.17958
.15121
.19732
.15751
.21464
.16392
S 47
.16170
.14511
.17988
.15131
.19761
.15762
.21493
.16403
/2 48
.16201
.14521
.18018
.15142
.19790
.15773
.21521
.16414
16 49
.16232
.14532
.18048
.15152
.19820
.15783
.21550
.16425
20 50
9.16262
0.14542
9.18077
0.15163
9.19849
0.15794
9.21578
0.16436
24 51
.16293
.14552
.18107
.15173
.19878
.15804
.21607
.16446
28 52
.16324
.14562
.18137
.15183
.19907
.15815
.21635
.16457
32 53
.16354
.14573
.18167
.15194
.19936
.15826
.21664
.16468
36 54
.16385
.14583
.18197
.15204
.19965
.15836
.21692
.16479
40 55
9.16415
0.14593
9.18227
0.15215
9.19995
0.15847
9.21721
0.16489
44 56
.16446
.14604
.18256
.15225
.20024
.15858
.21749
.16500
48 57
.16476
.14614
.18286
.15236
.20053
.15868
.21778
.16511
52 58
.16507
.14624
.18316
.15246
.20082
.15879
.21806
.16522
56 59
.16537
.14634
.18346
.15257
.20111
.15889
.21834
.16533
60 60
9.16568
0.14645
9.18376
0.15267
9.20140
0.15900
9.21863
0.16543
262
Table 10. Haversine Table
s '
3h 12m 48°
3* 16™ 49°
3>> 20m 50°
3* 24m 51°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.21863
0.16543
9.23545
0.17197
9.25190
0.17861
9.26797
0.18534
4 l
.21891
.16554
.23573
.17208
.25217
.17872
.26823
.18545
8 2
.21919
.16565
.23601
.17219
.25244
.17883
.26850
.18557
12 3
.21948
.16576
.23629
.17230
.25271
.17894
.26876
.18568
16 4
.21976
.16587
.23656
.17241
.25298
.17905
.26903
.18579
20 5
9.22004
0.16598
9.23684
0.17252
9.25325
0.17916
9.26929
0.18591
24 6
.22033
.16608
.23712
.17263
.25352
.17928
.26956
.18602
28 7
.22061
.16619
.23739
.17274
.25379
.17939
.26982
.18613
32 8
.22089
.16630
.23767
.17285
.25406
.17950
.27008
.18624
36 9
.22118
.16641
.23794
.17296
.25433
.17961
.27035
.18636
40 10
9.22146
0.16652
9.23822
0.17307
9.25460
0.17972
9.27061
0.18647
44 11
.22174
.16663
.23850
.17318
.25487
.17983
.27088
.18658
48 12
.22202
.16673
.23877
.17329
.25514
.17995
.27114
.18670
52 13
.22231
.16684
.23905
.17340
.25541
.18006
.27140
.18681
56 14
.22259
.16695
.23932
.17351
.25568
.18017
.27167
.18692
s '
3* 13™ 48°
3h 17m 49°
3h 21m 50°
3^ 25m 51°
0 15
9.22287
0.16706
9.23960
0.17362
9.25595
0.18028
9.27193
0.18704
4 16
.22315
.16717
.23988
.17373
.25622
.18039
.27219
.18715
S 17
.22343
.16728
.24015
.17384
.25649
.18050
.27246
.18727
12 18
.22372
.16738
.24043
.17395
.25676
.18062
.27272
.18738
J<? 19
.22400
.16749
.24070
.17406
.25703
.18073
.27298
.18749
20 20
9.22428
0.16760
9.24098
0.17417
9.25729
0.18084
9.27325
0.18761
24 21
.22456
.16771
.24125
.17428
.25756
.18095
.27351
.18772
25 22
.22484
.16782
.24153
.17439
.25783
.18106
.27377
.18783
32 23
.22512
.16793
.24180
.17450
.25810
.18118
.27403
.18795
3£ 24
.22540
.16804
.24208
.17461
.25837
.18129
.27430
.18806
40 25
9.22569
0.16815
9.24235
0.17472
9.25864
0.18140
9.27456
0.18817
44 26
.22597
.16825
.24263
.17483
.25891
.18151
.27482
.18829
45 27
.22625
.16836
.24290
.17494
.25917
.18162
.27508
.18840
52 28
.22653
.16847
.24317
.17505
.25944
.18174
.27535
.18852
5£ 29
.22681
.16858
.24345
.17517
.25971
.18185
.27561
.18863
s '
3h l^m 48°
3h is™ 49°
3h 2%m 50°
3h 26™ 51°
0 30
9.22709
0.16869
9.24372
0.17528
9.25998
0.18196
9.27587
0.18874
4 31
.22737
.16880
.24400
.17539
.26025
.18207
.27613
.18886
8 32
.22765
.16891
.24427
.17550
.26051
.18219
.27639
.18897
^2 33
.22793
.16902
.24454
.17561
.26078
.18230
.27666
.18908
Iff 34
.22821
.16913
.24482
.17572
.26105
.18241
.27692
.18920
20 35
9.22849
0.16924
9.24509
0.17583
9.26132
0.18252
9.27718
0.18931
24 36
.22877
.16934
.24536
.17594
.26158
.18263
.27744
.18943
28 37
.22905
.16945
.24564
.17605
.26185
.18275
.27770
.18954
32 38
.22933
.16956
.24591
.17616
.26212
.18286
.27796
.18965
3£ 39
.22961
.16967
.24618
.17627
.26238
.18297
.27822
.18977
40 40
9.22989
0.16978
9.24646
0.17638
9.26265
0.18308
9.27848
0.18988
44 41
.23017
.16989
.24673
.17649
.26292
.18320
.27875
.19000
48 42
.23045
.17000
.24700
.17661
.26319
.18331
.27901
.19011
52 43
.23073
.17011
.24728
.17672
.26345
.18342
.27927
.19022
5£ 44
.23100
.17022
.24755
.17683
.26372
.18353
.27953
.19034
s '
3* 15m 48°
gh 10m 49°
Sh 23™ 50°
3h 27m 51 u
0 45
9.23128
0.17033
9.24782
0.17694
9.26398
0.18365
9.27979
0.19045
4 46
.23156
.17044
.24809
.17705
.26425
.18376
.28005
.19057
S 47
.23184
.17055
.24837
.17716
.26452
.18387
.28031
.19068
J2 48
.23212
.17066
.24864
.17727
.26478
.18399
.28057
.19080
16 49
.23240
.17076
.24891
.17738
.26505
.18410
.28083
.19091
20 50
9.23268
0.17087
9.24918
0.17749
9.26532
0.18421
9.28109
0.19102
24 51
.23295
.17098
.24945
.17760
.26558
.18432
.28135
.19114
28 52
.23323
.17109
.24973
.17772
.26585
.18444
.28161
.19125
32 53
.23351
.17120
.25000
.17783
.26611
.18455
.28187
.19137
36 54
.23379
.17131
.25027
.17794
.26638
.18466
.28213
.19148
40 55
9.23407
0.17142
9.25054
0.17805
9.26664
0.18478
9.28239
0.19160
44 56
.23434
.17153
.25081
.17816
.26691
.18489
.28265
.19171
48 57
.23462
.17164
.25108
.17827
.26717
.18500
.28291
.19183
52 58
.23490
.17175
.25135
.17838
.26744
.18511
.28317
.19194
56 59
.23518
.17186
.25163
.17849
.26770
.18523
.28342
.19205
50 60
9.23545
0.17197
9.25190
0.17861
9.26797
0.18534
9.28368
0.19217
Table 10. Haversine Table
263
s '
Sh 28™ 52°
3* 32m 53°
Sh 36™ 54°
3* 40m 55°
Bvr.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
).2S3(iS
0.19217
9.29906
0.19909
9.31409
0.20611
9.32881
021321
4 1
.28394
.19228
.29931
.19921
.31434
.20623
.32905
.21333
8 2
.28420
.19240
.29956
.19932
.31459
.20634
.32930
.21345
12 3
.28446
.19251
.29981
.19944
.31484
.20646
.32954
.21357
16 4
.28472
.19263
.30007
.19956
.31508
.20658
.32978
.21369
20 5
9.28498
0.19274
9.30032
0.19967
9.31533
0.20670
9.33002
0.21381
24 6
.28524
.19286
.30057
.19979
.31558
.20681
.33027
.21393
28 7
.28549
.19297
.30083
.19991
.31583
.20693
.33051
.21405
32 8
.28575
.19309
.30108
.20002
.31607
.20705
.33075
.21417
86 9
.28601
.19320
.30133
.20014
.31632
.20717
.33099
.21429
40 10
9.28627
0.19332
9.30158
0.20026
9.31657
0.20729
9.33123
0.21440
-a 11
.28653
.19343
.30184
.20037
.31682
.20740
.33148
.21452
48 12
.28679
.19355
.30209
.20049
.31706
.20752
.33172
.21464
52 13
.28704
.19366
.30234
.20060
.31731
.20764
.33196
.21476
56 14
.28730
.19378
.30259
.20072
.31756
.20776
.33220
.21488
s '
3h 2ST 52°
3h 33™ 53°
Sh 37^ 54°
3h 41m 55°
0 15
9.28756
0.19389
9.30285
0.20084
9.31780
0.20788
9.33244
0.21500
4 16
.28782
.19401
.30310
.20095
.31805
.20799
.33268
.21512
S 17
.28807
.19412
.30335
.20107
.31830
.20811
.33292
.21524
12 18
.28833
.19424
.30360
.20119
.31854
.20823
.33317
.21536
J<? 19
.28859
.19435
.30385
.20130
.31879
.20835
.33341
.21548
20 20
9.28885
0.19447
9.30410
0.20142
9.31903
0.20847
9.33365
0.21560
24 21
.28910
.19458
.30436
.20154
.31928
.20858
.33389
.21572
25 22
.28936
.19470
.30461
.20165
.31953
.20870
.33413
.21584
32 23
.28962
.19481
.30486
.20177
.31977
.20882
.33437
.21596
35 24
.28987
.19493
.30511
.20189
.32002
.20894
.33461
.21608
40 25
9.29013
0.19504
9.30536
0.20200
9.32026
0.20906
9.33485
0.21620
44 26
.29039
.19516
.30561
.20212
.32051
.20918
.33509
.21632
45 27
.29064
.19527
.30586
.20224
.32076
.20929
.33533
.21644
52 28
.29090
.19539
.30611
.20235
.32100
.20941
.33557
.21656
55 29
.29116
.19550
.30636
.20247
.32125
.20953
.33581
.21668
s '
3* 30m 52°
gh 3jm 53°
Sh 38^ 54°
3h 42"' 55°
0 30
9.29141
0.19562
9.30662
0.20259
9.32149
0.20965
9.33605
0.21680
4 31
.29167
.19573
.30687
.20271
.32174
.20977
.33629
.21692
8 32
.29192
.19585
.30712
.20282
.32198
.20989
.33653
.21704
72 33
.29218
.19597
.30737
.20294
.32223
.21000
.33677
.21716
76 34
.29244
.19608
.30762
.20306
.32247
.21012
.33701
.21728
20 35
9.29269
0.19620
9.30787
0.20317
9.32272
0.21024
9.33725
0.21740
24 36
.29295
.19631
.30812
.20329
.32296
.21036
.33749
.21752
28 37
.29320
.19643
.30837
.20341
.32321
.21048
.33773
.21764
32 38
.29346
.19654
.30862
.20352
.32345
.21060
.33797
.21776
36 39
.29371
.19666
.30887
.20364
.32370
.21072
.33821
.21788
40 40
9.29397
0.19677
9.30912
0.20376
9.32394
0.21083
9.33845
0.21800
44 41
.29422
.19689
.30937
.20388
.32418
.21095
.33869
.21812
48 42
.29448
.19701
.30962
.20399
.32443
.21107
.33893
.21824
52 43
.29473
.19712
.30987
.20411
.32467
.21119
.33917
.21836
55 44
.29499
.19724
.31012
.20423
.32492
.21131
.33941
.21848
s '
3h sim 52°
3* 35m 53°
3h 39™ 54°
Sh 43" 55°
0 45
9.29524
0.19735
9.31036
0.20435
9.32516
0.21143
9.33965
0.21860
4 46
.29550
.19747
.31061
.20446
.32541
.21155
.33988
.21872
S 47
.29575
.19758
.31086
.20458
.32565
.21167
.34012
.21884
12 48
.29601
.19770
.31111
.20470
.32589
.21178
.34036
.21896
16 49
.29626
.19782
.31136
.20481
.32614
.21190
.34060
.21908
20 50
9.29652
0.19793
9.31161
0.20493
9.32638
0.21202
9.34084
0.21920
24 51
.29677
.19805
.31186
.20505
.32662
.21214
.34108
.21932
28 52
.29703
.19816
.31211
.20517
.32687
.21226
.34132
.21944
32 53
.29728
.19828
.31236
.20528
.32711
.21238
.34155
.21956
36 54
.29753
.19840
.31260
.20540
.32735
.21250
.34179
.21968
40 55
9.29779
0.19851
9.31285
0.20552
9.32760
0.21262
9.34203
0.21980
44 56
.29804
.19863
.31310
.20564
.32784
.21274
.34227
.21992
48 57
.29829
.19874
.31335
.20575
.32808
.21285
.34251
.22004
52 58
.29855
.19886
.31360
.20587
.32833
.21297
.34274
.22016
56 59
.29880
.19898
.31385
.20599
.32857
.21309
.34298
.22028
(SO 60
9.29906
0.19909
9.31409
0.20611
9.32881
0.21321
9.34322
0.22040
264
Table 10. Haversine Table
,
SA 44m 56°
gA 48™ 57°
3* 52™ 58°
3* 56™ 59°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.34322
0.22040
9.35733
0.22768
9.37114
0.23504
9.38468
0.24248
4 1
.34346
.22052
.35756
.22780
.37137
.23516
.38490
.24261
8 2
.34369
.22064
.35779
.22792
.37160
.23529
.38512
.24273
12 3
.34393
.22077
.35802
.22805
.37183
.23541
.38535
.24286
16 4
.34417
.22089
.35826
.22817
.37205
.23553
.38557
.24298
20 5
9.34441
0.22101
9.35849
0.22829
9.37228
0.23566
9.38579
0.24310
24 6
.34464
.22113
.35872
.22841
.37251
.23578
.38602
.24323
28 1
.34488
.22125
.35895
.22853
.37274
.23590
.38624
.24335
32 8
.34512
.22137
.35918
.22866
.37296
.23603
.38646
.24348
Sff 9
.34535
.22149
.35942
.22878
.37319
.23615
.38668
.24360
40 10
9.34559
0.22161
9.35965
0.22890
9.37342
0.23627
9.38691
0.24373
44 11
.34583
.22173
.35988
.22902
.37364
.23640
.38713
.24385
48 12
.34606
.22185
.36011
.22915
.37387
.23652
.38735
.24398
52 13
.34630
.22197
.36034
.22927
.37410
.23665
.38757
.24410
off 14
.34654
.22209
.36058
.22939
.37433
.23677
.38780
.24423
s '
3* 45m 56°
3h 49m 57°
3* 5S"1 58°
3h 57« 59°
0 15
9.34677
0.22221
9.36081
0.22951
9.37455
0.23689
9.38802
0.24435
4 16
.34701
.22234
.36104
.22964
.37478
.23702
.38824
.24448
5 17
.34725
.22246
.36127
.22976
.37501
.23714
.38846
.24460
12 18
.34748
.22258
.36150
.22988
.37523
.23726
.38868
.24473
16 19
.34772
.22270
.36173
.23000
.37546
.23739
.38891
.24485
20 20
9.34795
0.22282
9.36196
0.23012
9.37569
0.23751
9.38913
0.24498
24. 21
.34819
.22294
.36219
.23025
.37591
.23764
.38935
.24510
25 22
.34843
.22306
.36243
.23037
.37614
.23776
.38957
.24523
32 23
.34866
.22318
.36266
.23049
.37636
.23788
.38979
.24535
36 24
.34890
.22330
.36289
.23061
.37659
.23801
.39002
.24548
40 25
9.34913
0.22343
9.36312
0.23074
9.37682
0.23813
9.39024
0.24560
44 26
.34937
.22355
.36335
.23086
.37704
.23825
.39046
.24573
48 27
.34960
.22367
.36358
.23098
.37727
.23838
.39068
.24586
52 25
.34984
.22379
.36381
.23110
.37749
.23850
.39090
.24598
56 29
.35007
.22391
.36404
.23123
.37772
.23863
.39112
.24611
s '
3* 46™ 56°
3h 50m 57°
3* 54m 58°
3h 58™ 59°
0 30
9.35031
0.22403
9.36427
0.23135
9.37794
0.23875
9.39134
0.24623
4 31
.35054
.22415
.36450
.23147
.37817
.23887
.39156
.24636
5 32
.35078
.22427
.30473
.23160
.37840
.23900
.39178
.24648
12 33
.35101
.22440
.36496
.23172
.37862
.23912
.39201
24661
J6 34
.35125
.22452
.36519
.23184
.37885
.23925
.39223
.24673
20 35
9.35148
0.22464
9.36542
0.23196
9.37907
0.23937
9.39245
0.24686
24 36
.35172
.22476
.36565
.23209
.37930
.23950
.39267
.24698
25 37
.35195
.22488
.36588
.23221
.37952
.23962
.39289
.24711
32 38
.35219
.22500
.36611
.23233
.37975
.23974
.39311
.24723
36 39
.35242
.22512
.36634
.23246
.37997
.23987
.39333
.24736
40 40
9.35266
0.22525
9.36657
0.23258
9.38020
0.23999
9.39355
0.24749
44 41
.35289
.22537
.36680
.23270
.38042
.24012
.39377
.24761
45 42
.35312
.22549
.36703
.23282
.38065
.24024
.39399
.24774
52 43
.35336
.22561
.36726
.23295
.38087
.24036
.39421
.24786
56 44
.35359
.22573
.36749
.23307
.38110
.24049
39443
.24799
s '
3* 47m 56°
3h 51m 57°
3h 55m 58°
3h SQ™ 59°
0 45
9.35383
0.22585
9.36772
0.23319
9.38132
0.24061
9.39465
0.24811
4 46
.35406
.22598
.36794
.23332
.38154
.24074
.39487
.24824
5 47
.35429
.22610
.36817
.23344
.38177
.24086
.39509
.24836
12 48
.35453
.22622
.36840
.23356
.38199
.24099
.39531
.24849
16 49
.35476
.22634
.36863
.23368
.38222
.24111
.39553
.24862
20 50
9.35500
0.22646
9.36886
0.23381
9.38244
0.24124
9.39575
0.24874
24 51
.35523
.22658
.36909
.23393
.38267
.24136
.39597
.24887
25 52
.35546
.22671
.36932
.23405
.38289
.24148
.39619
.24899
32 53
.35570
.22683
.36955
.23418
.38311
.24161
.39641
.24912
36 54
.35593
.22695
.36977
.23430
.38334
.24173
.39663
.24924
40 55
9.35616
0.22707
9.37000
0.23442
9.38356
0.24186
9.39685
0.24937
44 56
.35639
.22719
.37023
.23455
.38378
.24198
.39706
.24950
45 57
.35663
.22731
.37046
.23467
.38401
.24211
.39728
.24962
52 58
.35686
.22744
.37069
.23479
.38423
.24223
.39750
.24975
56 59
.35709
.22756
.37091
.23492
.38445
.24236
.39772
.24987
50 60
9.35733
0.22768
9.37114
0.235C4
9.38468
0.24248
9.39794
0.25000
Table 10. Hayersine Table
265
s '
4* Om 60°
4h 4m 61°
4* 8m 62°
4* 12™ 63°
Hav.
No.
Uav.
No.
Hav.
No.
Hav.
No.
0 0
9.39794
0.25000
9.41094
0.25760
9.42368
0.26526
9.43617
0.27300
4 1
.39816
.25013
.41115
.25772
.42389
.26539
.43638
.27313
8 2
.39838
.25025
.41137
.25785
.42410
.26552
.43658
.27326
12 3
.39860
.25038
.41158
.25798
.42431
.26565
.43679
.27339
16 4
.39881
.25050
.41180
.25810
.42452
.26578
.43699
.27352
20 5
9.39903
0.25063
9.41201
0.25823
9.42473
0.26591
9.43720
0.27365
24 6
.39925
.25076
.41222
.25836
.42494
.26604
.43741
.27378
28 7
.39947
.25088
.41244
.25849
.42515
.26616
.43761
.27391
32 8
.39969
.25101
.41265
.25861
.42536
.26629
.43782
.27404
36 9
.39991
.25113
.41287
.25874
.42557
.26642
.43802
.27417
40 10
9.40012
0.25126
9.41308
0.25887
9.42578
0.26655
9.43823
0.27430
44 11
.40034
.25139
.41329
.25900
.42599
.26668
.43843
.27443
48 12
.40056
.25151
.41351
.25912
.42620
.26681
.43864
.27456
52 13
.40078
.25164
.41372
.25925
.42641
.26694
.43884
.27469
56 14
.40100
.25177
.41393
.25938
.42662
.26706
.43905
.27482
s '
4* lm 60°
4h 5m 61°
4* 9m 62°
4* 13m 63°
0 15
9.40121
0.25189
9.41415
0.25951
9.42682
0.26719
9.43926
0.27495
4 16
.40143
.25202
.41436
.25963
.42703
.26732
.43946
.27508
S 17
.40165
.25214
.41457
.25976
.42724
.26745
.43967
.27521
12 18
.40187
.25227
.41479
.25989
.42745
.26758
.43987
.27534
/<? 19
.40208
.25240
.41500
.26002
.42766
.26771
.44008
.27547
20 20
9.40230
0.25252
9.41521
0.26014
9.42787
0.26784
9.44028
0.27560
24 21
.40252
.25265
.41543
.26027
.42808
.26797
.44048
.27573
2<S 22
.40274
.25278
.41564
.26040
.42829
.26809
.44069
.27586
32 23
.40295
.25290
.41585
.26053
.42850
.26822
.44089
.27599
36 24
.40317
.25303
.41606
.26065
.42870
.26835
.44110
.27612
40 25
9.40339
0.25316
9.41628
0.26078
9.42891
0.26848
9.44130
0.27625
44 26
.40360
.25328
.41649
.26091
.42912
.26861
.44151
.27638
45 27
.40382
.25341
.41670
.26104
.42933
.26874
.44171
.27651
52 28
.40404
.25354
.41692
.26117
.42954
.26887
.44192
.27664
56 29
.40425
.25366
.41713
.26129
.42975
.26900
.44212
.27677
s '
4* 2™ 60°
4» 6™ 61°
4k 10™ 62°
4* I4m 63°
0 30
9.40447
0.25379
9.41734
0.26142
9.42996
0.26913
9.44232
0.27690
4 31
.40469
.25391
.41755
.26155
.43016
.26925
.44253
.27703
8 32
.40490
.25404
.41776
.26168
.43037
.26938
.44273
.27716
J2 33
.40512
.25417
.41798
.26180
.43058
.26951
.44294
.27729
/'/ 34
.40534
.25429
.41819
.26193
.43079
.26964
.44314
.27742
20 35
9.40555
0.25442
9.41840
0.26206
9.43100
0.26977
9.44334
0.27755
24 36
.40577
.25455
.41861
.26219
.43120
.26990
.44355
.27768
28 37
.40599
.25467
.41882
.26232
.43141
.27003
.44375
.27781
32 38
.40620
.25480
.41904
.26244
.43162
.27016
.44396
.27794
36 39
.40642
.25493
.41925
.26257
.43183
.27029
.44416
.27807
40 40
9.40663
0.25506
9.41946
0.26270
9.43203
0.27042
9.44436
0.27820
44 41
.40685
.25518
.41967
.26283
.43224
.27055
.44457
.27833
48 42
.40707
.25531
.41988
.26296
.43245
.27068
.44477
.27846
52 43
.40728
.25544
.42009
26308
.43266
.27080
.44497
.27859
56 44
.40750
.25556
.42031
.26321
.43286
.27093
.44518
.27873
8 '
4*3- 60°
4* 7m 61°
4* llm 62°
4* I5m 63°
0 45
9.40771
0.25569
9.42052
0.26334
9.43307
0.27106
9.44538
0.27886
4 46
.40793
.25582
.42073
.26347
.43328
.27119
.44558
.27899
S 47
.40814
.25594
.42094
26360
.43348
.27132
.44579
.27912
/2 48
.40836
.25607
.42115
.26372
.43369
.27145
.44599
.27925
16 49
.40858
.25620
.42136
.26385
.43390
.27158
.44619
.27938
20 50
9.40879
025632
9.42157
0.26398
9.43411
0.27171
9.44639
0.27951
24 51
.40900
.25645
.42178
.26411
.43431
.27184
.44660
.27964
2S 52
.40922
.25658
.42199
.26424
.43452
.27197
.44680
.27977
32 63
.40943
.25671
.42221
.26437
.43473
.27210
.44700
.27990
36 54
.40965
.25683
.42242
.26449
.43493
.27223
.44721
.28003
40 55
9.40986
0.25696
9.42263
0.26462
9.43514
0.27236
9.44741
0.28016
44 66
.41008
.25709
.42284
.26475
.43535
.27249
.44761
.28029
48 57
.41029
.25721
.42305
.26488
.43555
.27262
.44781
.28042
52 58
.41051
.25734
.42326
.26501
.43576
.27275
.44801
.28055
->H 59
.41072
.25747
.42347
.26514
.43596
.27288
.44822
.28068
«rt f,c.
Q 4 1 HO/!
n 9R7Afl
Q /lOQftC
n QCCOC
O AtRf7
n OTinn
1 1 1 v !•>
n 9«n«i
266
Table 10. Haversine Table
s '
4* K?"1 64°
4* 20m 65°
4* 24m . 66°
4h 28™ 67°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.44842
0.28081
9.46043
0.28869
9.47222
0.29663
9.48378
0.30463
4 1
.44862
.28095
.46063
.28882
.47241
.29676
.48397
.30477
8 2
.44882
.28108
.46083
.28895
.47261
.29690
.48416
.30490
12 3
.44903
.28121
.46103
.28909
.47280
.29703
.48435
.30504
16 4
.44923
.28134
.46123
.28922
.47300
.29716
.48454
.30517
20 5
9.44943
0.28147
9.46142
0.28935
9.47319
0.29730
9.48473
0.30530
24 6
.44963
.28160
.46162
.28948
.47338
.29743
.48492
.30544
28 7
.44983
.28173
.46182
.28961
.47358
.29756
.48511
.30557
32 8
.45003
.28186
.46202
.28975
.47377
.29770
.48530
.30571
36 9
.45024
.28199
.46222
.28988
.47397
.29783
.48549
.30584
40 10
9.45044
0.28212
9.46241
0.29001
9.47416
0.29796
9.48568
0.30597
44 11
.45064
.28225
.46261
.29014
.47435
.29809
.48587
.30611
48 12
.45084
.28238
.46281
.29027
.47455
.29823
.48607
.30624
52 13
.45104
.28252
.46301
.29041
.47474
.29836
.48626
.30638
56 14
.45124
.28265
.46320
.29054
.47493
.29849
.48645
.30651
s '
4* 17m 64°
4h 21m 65°
4* 25'" 66°
4* 29™ 67°
0 15
9.45144
0.28278
9.46340
0.29067
9.47513
0.29863
9.48664
0.30664
4 16
.45165
.28291
.46360
.29080
.47532
.29876
.48683
.30678
S 17
.45185
.28304
.46380
.29093
.47552
.29889
.48702
.30691
12 18
.45205
.28317
.46399
.29107
.47571
.29903
.48720
.30705
1£ 19
.45225
.28330
.46419
.29120
.47590
.29916
.48739
.30718
20 20
9.45245
0.28343
9.46439
0.29133
9.47610
0.29929
9.48758
0.30732
24 21
.45265
.28356
.46458
.29146
.47629
.29943
.48777
.30745
25 22
.45285
.28369
.46478
.29160
.47648
.29956
.48796
.30758
32 23
.45305
.28383
.46498
.29173
.47668
.29969
.48815
.30772
3£ 24
.45325
.28396
.46517
.29186
.47687
.29983
.48834
.30785
40 25
9.45345
0.28409
9.46537
0.29199
9.47706
0.29996
9.48853
0.30799
44 26
.45365
.28422
.46557
.29212
.47725
.30009
.48872
.30812
45 27
.45385
.28435
.46576
.29226
.47745
.30023
.48891
.30826
52 28
.45405
.28448
.46596
.29239
.47764
.30036
.48910
.30839
56 29
.45426
.28461
.46616
.29252
.47783
.30049
.48929
.30852
s '
4* 18™ 64°
4h 22™ 65°
4* 26™ 66°
4* SO™ 67°
0 30
9.45446
0.28474
9.46635
0.29265
9.47803
0.30063
9.48948
0.30866
4 31
.45466
.28488
.46655
.29279
.47822
.30076
.48967
.30879
8 32
.45486
.28501
.46675
.29292
.47841
.30089
.48986
.30893
12 33
.45506
.28514
.46694
.29305
.47860
.30103
.49004
.30906
1<S 34
.45526
.28527
.46714
.29318
.47880
.30116
.49023
.30920
20 35
9.45546
0.28540
9.46733
0.29332
9.47899
0.30129
9.49042
0.30933
24 36
.45566
.28553
.46753
.29345
.47918
.30143
.49061
.30946
28 37
.45586
.28566
.46773
.29358
.47937
.30156
•49080
.30960
32 38
.45606
.28580
.46792
.29371
.47957
.30169
.49099
.30973
36 39
.45625
.28593
.46812
.29385
.47976
.30183
.49118
.30987
40 40
9.45645
0.28606
9.46831
0.29398
9.47995
0.30196
9.49137
0.31000
44 41
.45665
.28619
.46851
.29411
.48014
.30209
.49155
.31014
48 42
.45685
.28632
.46871
.29424
.48033
.30223
.49174
.31027
52 43
.45705
.28645
.46890
.29438
.48053
.30236
.49193
.31041
56 44
.45725
.28658
.46910
.29451
.48072
.30249
£9212
.31054
s '
4* 19™ 64°
4* 23™ 65°
4* 27™ 66°
4* 31™ 67°
0 45
9.45745
0.28672
9.46929
0.29464
9.48091
0.30263
9.49231
0.31068
4 46
.45765
.28685
.46949
.29477
.48110
.30276
.49250
.31081
5 47
.45785
.28698
.46968
.29491
.48129
.30290
.49268
.31095
12 48
.45805
.28711
.46988
.29504
.48148
.30303
.49287
.31108
16 49
.45825
.28724
.47007
.29517
.48168
.30316
.49306
.31121
20 50
9.45845
0.28737
9.47027
0.29530
9.48187
0.30330
9.49325
0.31135
24 51
.45865
.28751
.47046
.29544
.48206
.30343
.49344
.31148
28 52
.45884
.28764
.47066
.29557
.48225
.30356
.49362
.31162
32 53
.45904
.28777
.47085
.29570
.48244
.30370
.49481
.31175
36 54
.45924
.28790
.47105
.29583
.48263
.30383
.49400
.31189
40 55
9.45944
0.28803
9.47124
0.29597
9.48282
0.30397
9.49419
0.31202
44 56
.45964
.28816
.47144
.29610
.48302
.30410
.49437
.31216
48 57
.45984
.28830
.47163
.29623
.48321
.30423
.49456
.31229
52 58
.46004
.28843
,47183
.29637
.48340
.30437
.49475
.31243
56 59
.46023
.28856
.47202
.29650
.48359
.30450
.49494
.31256
60 60
9.46043
0.28869
9.47222
0.29663
9.48378
0.30463
9.49512
0.31270
Table 10. Haversine Table
26'
s '
4* S2m 68°
4* 36™ 69°
4* 4Qm 70°
4* 44m 71°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.49512
0.31270
9.50626
0.32082
9.51718
0.32899
9.52791
0.33722
4 1
.49531
.31283
.50644
.32095
.51736
.32913
.52809
.33735
8 2
.49550
.31297
.50662
.32109
.51754
.32926
.52826
.33749
12 3
.49568
.31310
.50681
.32122
.51772
.32940
.52844
.33763
16 4
.49587
.31324
.50699
.32136
.51790
.32954
.52862
.33777
20 5
9.49606
0.31337
9.50717
0.32150
9.51808
0.32967
9.52879
0.33790
24 6
.49625
.31351
.50736
.32163
.51826
.32981
.52897
.33804
28 7
.49643
.31364
.50754
.32177
.51844
.32995
.52915
.33818
32 8
.49662
.31378
.50772
.32190
.51862
.33008
.52932
.33832
36 9
.49681
.31391
.50791
.32204
.51880
.33022
.52950
.33845
40. 10
9.49699
0.31405
9.50809
0.32217
9.51898
0.33036
9.52968
0.33859
44 11
.49718
.31418
.50827
.32231
.51916
.33049
.52985
.33873
48 12
.49737
.31432
.50846
.32245
.51934
.33063
.53003
.33887
52 13
.49755
.31445
.50864
.32258
.51952
.33077
.53021
.33900
56 14
.49774
.31459
.50882
.32272
.51970
.33090
.53038
.33914
s '
4* 33"' 68°
4* 37m 69°
4* 41m 70°
4* 45m 71°
0 15
9.49793
0.31472
9.50901
0.32285
9.51988
0.33104
9.53056
0.33928
4 16
.49811
.31486
.50919
.32299
.52006
.33118
.53073
.33942
S 17
.49830
.31499
.50937
.32313
.52024
.33132
.53091
.33956
12 18
.49849
.31513
.50956
.32326
.52042
.33145
.53109
.33969
76 19
.49867
.31526
.50974
.32340
.52060
.33159
.53126
.33983
20 20
9.49886
0.31540
9.50992
0.32353
9.52078
0.33173
9.53144
0.33997
24 21
.49904
.31553
.51010
.32367
.52096
.33186
.53162
.34011
25 22
.49923
.31567
.51029
.32381
.52114
.33200
.53179
.34024
32 23
.49942
.31580
.51047
.32394
.52132
.33214
.53197
.34038
36 24
.49960
.31594
.51065
.32408
.52150
.33227
.53214
.34052
40 25
9.49979
0.31607
9.51083
0.32422
9.52168
0.33241
9.53232
0.34066
44 26
.49997
.31621
.51102
.32435
.52185
.33255
.53249
.34080
4S 27
.50016
.31634
.51120
.32449
.52203
.33269
.53267
.34093
.52 28
.50034
.31648
.51138
.32462
.52221
.33282
.53285
.34107
5(5 29
.50053
.31661
.51156
.32476
.52239
.33296
.53302
.34121
s '
4* S4m 68°
4* 38™ 69°
4* 42m 70°
4* 46m 71°
0 30
9.50072
0.31675
9.51174
0.32490
9.52257
0.33310
9.53320
0.34135
^ 31
.50090
.31688
.51193
.32503
.52275
.33323
.53337
.34149
8 32
.50109
.31702
.51211
.32517
.52293
.33337
.53355
.34162
/2 33
.50127
.31716
.51229
.32531
.52311
.33351
.53372
.34176
76 34
.50146
.31729
.51247
.32544
.52328
.33365
.53390
.34190
20 35
9.50164
0.31742
9.51265
0.32558
9.52346
0.33378
9.53407
0.34204
24 36
.50183
.31756
.51284
.32571
.52364
.33392
.53425
.34218
28 37
.50201
.31770
.51302
.32585
.52382
.33406
.53442
.34231
32 38
.50220
.31783
.51320
.32599
.52400
.33419
.53460
.34245
36 39
.50238
.31797
.51338
.32612
.52418
.33433
.53477
.34259
40 40
9.50257
0.31810
9.51356
032626
9.52436
0.33447
9.53495
0.34273
-44 41
.50275
.31824
.51374
.32640
.52453
.33461
.53512
.34287
48 42
.50294
.31837
.51393
.32653
.52471
.33474
.53530
.34300
52 43
.50312
.31851
.51411
.32667
.52489
.33488
.53547
.34314
56 44
.50331
.31865
.51429
.32681
.52507
.33502
.53565
.34328
8 '
4A 35m 68°
4* 30" 69°
4* 43" 70°
4* 47" 71°
0 45
9.50349
0.31878
9.51447
0.32694
9.52525
0.33515
9.53582
0.34342
4 46
.50368
.31892
.51465
.32708
.52542
.33529
.53600
.34356
5 47
.50386
.31905
.51483
.32721
.52560
.33543
.53617
.34369
12 48
.50405
.31919
.51501
.32735
.52578
.33557
.53635
.34383
16 49
.50423
.31932
.51519
.32749
.52596
.33570
.53652
.34397
20 50
9.50442
0.31946
9.51538
0.32762
9.52613
0.33584
9.53670
0.34411
24 51
.50460
.31959
.51556
.32776
.52631
.33598
.53687
.34425
28 52
.50478
.31973
.51574
.32790
.52649
.33612
.53704
.34439
32 53
.50497
.31987
.51592
.32803
.52667
.33625
.53722
.34452
36 54
.50515
.32000
.51610
.32817
.52684
.33639
.53739
.34466
^0 55
9.50534
0.32014
9.51628
0.32831
9.52702
0.33653
9.53757
0.34480
44 56
.50552
.32027
.51646
.32844
.52720
.33667
.53774
.34494
4S 57
.50570
.32041
.51664
.32858
.52738
.33680
.53792
.34508
52 58
.50589
.32054
.51682
.32872
.52755
.33694
.53809
.34521
56 59
.50607
.32068
.51700
.32885
.52773
.33708
.53826
.34535
60 60
9.50626
0.32082
9.51718
0.32899
9.52791
0.33722
9.53844
0.34549
268
Table 10. Haversine Table
s '
4^ 45™ 72°
4* 52m 73°
4h 56m 74«
ffh Om 75°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.53844
0.34549
9.54878
0.35381
9.55893
0.36218
9.56889
0.37059
4 1
.53861
.34563
.54895
.35395
.55909
.36232
.56906
.37073
8 2
.53879
.34577
.54912
.35409
.55926
.36246
.56922
.37087
12 3
.53896
.34591
.54929
.35423
.55943
.36260
.56939
.37101
16 4
.53913
.34604
.54946
.35437
.55960
.36274
.56955
.37115
20 5
9.53931
0.34618
9.54963
0.35451
9.55976
0.36288
9.56972
0.37129
24 6
.53948
.34632
.54980
.35465
.55993
.36302
.56988
.37143
28 7
.53966
.34646
.54997
.35479
.56010
.36316
.57005
.37157
32 8
.53983
.34660
.55014
.35493
.56027
.36330
.57021
.37171
36 9
.54000
.34674
.55031
.35507
.56043
.36344
.57037
.37186
40 10
9.54017
0.34688
9.55048
0.35521
9.56060
0.36358
9.57054
0.37200
44 11
.54035
.34701
.55065
.35534
.56077
.36372
.57070
.37214
48 12
.54052
.34715
.55082
.35548
.56093
.36386
.57087
.37228
52 13
.54069
.34729
.55099
.35562
.56110
.36400
.57103
.37242
56 14
.54087
.34743
.55116
.35576
.56127
.36414
.57119
.37256
s '
4*. 4,9"' 72°
4* 53™ 73°
4* 57m 74°
gh lm 75°
0 15
9.54104
0.34757
9.55133
0.35590
9.56144
0.36428
9.57136
0.37270
4 16
.54121
.34771
.55150
.35604
.56160
.36442
.57152
.37284
5 17
.54139
.34784
.55167
.35618
.56177
.36456
.57169
.37298
12 18
.54156
.34798
.55184
.35632
.56194
.36470
.57185
.37312
iff 19
.54173
.34812
.55201
.35646
.56210
.36484
.57201
.37326
20 20
9.54190
0.34826
9.55218
0.35660
9.56227
0.36498
9.57218
0.37340
24 21
.54208
.34840
.55235
.35674
.56244
.36512
.57234
.37354
25 22
.54225
.34854
.55252
.35688
.56260
.36526
.57250
.37368
32 23
.54242
.34868
.55269
.35702
.56277
.36540
.57267
.37382
Sff 24
.54260
.34882
.55286
.35716
.56294
.36554
.57283
.37397
40 25
9.54277
0.34895
9.55303
0.35730
9.56310
0.36568
9.57299
0.37411
44 26
.54294
.34909
.55320
.35743
.56327
.36582
.57316
.37425
45 27
.54311
.34923
.55337
.35757
.56343
.36596
.57332
.37439
52 28
.54329
.34937
.55354
.35771
.56360
.36610
.57348
.37453
56 29
.54346
.34951
.55370
.35785
.56377
.36624
.57365
.37467
s '
4* 50m 72°
4h 54m 73°
4A 58™ 74°
5h 2m 75°
0 30
9.54363
0.34965
9.55387
0.35799
9.56393
0.36638
9.57381
0.37481
4 31
.54380
.34979
.55404
.35813
.56410
.36652
.57397
.37495
8 32
.54397
.34992
.55421
.35827
.56426
.36666
.57414
.37509
^2 33
.54415
.35006
.55438
.35841
.56443
.36680
.57430
.37523
iff 34
.54432
.35020
.55455
.35855
.56460
.36694
.57446
.37537
20 35
9.54449
0.35034
9.55472
0.35869
9.56476
0.36708
9.57463
0.37551
24 36
.54466
.35048
.55489
.35883
.56493
.36722
.57479
.37566
28 37
.54483
.35062
.55506
.35897
.56509
.36736
.57495
.37580
32 38
.54501
.35076
.55523
.35911
.56526
.36750
.57511
.37594
Sff 39
.54518
.35090
.55539
.35925
.56543
.36764
.57528
.37608
40 40
9.54535
0.35103
9.55556
0.35939
9.56559
0.36778
9.57544
0.37622
44 41
.54552
.35117
.55573
.35953
.56576
.36792
.57560
.37636
48 42
.54569
.35131
.55590
.35967
.56592
.36806
.57577
.37650
52 43
.54587
.35145
.55607
.35981
.56609
.36820
.57593
.37664
50 44
.54604
.35159
.55624
.35995
.56625
.36834
.57609
.37678
s '
4* 51 m 72°
4* 55m 73°
4* 59™ 74°
5h gm 75°
0 45
9.54621
0.35173
9.55641
0.36009
9.56642
0.36848
9.57625
0.37692
4 46
.54638
.35187
.55657
.36023
.56658
.36862
.57642
.37706
S 47
.54655
.35201
.55674
.36036
.56675
.36877
.57658
.37721
.72 48
.54672
.35215
.55691
.36050
.56692
.36891
.57674
.37735
16 49
.54689
.35228
.55708
.36064
.56708
.36905
.57690
.37749
20 50
9.54707
0.35242
9.55725
0.36078
9.56725
0.36919
9.57706
0.37763
24 51
.54724
.35256
.55742
.36092
.56741
.36933
.57723
.37777
28 52
.54741
.35270
.55758
.36106
.56758
.36947
.57739
.37791
32 53
.54758
.35284
.55775
.36120
.56774
.36961
.57755
.37805
36 54
.54775
.35298
.55792
.36134
.56791
.36975
.57771
.37819
40 55
9.54792
0.35312
9.55809
0.36148
9.56807
0.36989
9.57787
0.37833
44 56
.54809
.35326
.55826
.36162
.56824
.37003
.57804
.37847
48 57
.54826
.35340
.55842
.36176
.56840
.37017
.57820
.37862
52 58
.54843
.35354
.55859
.36190
.56856
.37031
.57836
.37876
56 59
.54860
.35368
.55876
.36204
.56873
.37045
.57852
.37890
60 60
9.54878
0.35381
9.55893
0.36218
9.56889
0.37059
9.57868
0.37904
Table 10. Haversine Table
269
s '
oh 4m 76°
gh gm 77°
5h 12™ 78°
fjh Iffn 79°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.57868
0.37904
9.58830
0.38752
9.59774
0.39604
9.60702
0.40460
4 1
.57885
.37918
.58846
.38767
.59790
.39619
.60717
.40474
8 2
.57901
.37932
.58862
.38781
.59806
.39633
.60733
.40488
12 3
.57917
.37946
.58878
.38795
.59821
.39647
.60748
.40502
16 4
.57933
.37960
.58893
.38809
.59837
.39661
.60763
.40517
20 5
9.57949
0.37974
9.58909
0.38823
9.59852
0.39676
9.60779
0.40531
24 6
.57965
.37989
.58925
.38837
.59868
.39690
.60794
.40545
28 7
.57981
.38003
.58941
.38852
.59883
.39704
.60809
.40560
32 8
.57998
.38017
.58957
.38866
.59899
.39718
.60825
.40574
36 9
.58014
.38031
.58973
.38880
.59915
.39732
.60840
.40588
40 10
9.58030
0.38045
9.58989
0.38894
9.59930
0.39746
9.60855
0.40602
44 11
.58046
.38059
.59004
.38908
.59946
.39761
.60870
.40617
48 12
.58062
.38073
.59020
.38923
.59961
.39775
.60886
.40631
5£ 13
.58078
.38087
.59036
.38937
.59977
.39789
.60901
.40645
56 14
.58094
.38102
.59052
.38951
.59992
.39803
.60916
.40660
s '
5h 5m 76°
gh Qm 77°
5h IS"1 78°
5h !Jm 790
0 15
9.58110
0.38116
9.59068
0.38965
9.60008
0.39818
9.60931
0.40674
4 16
.58126
.38130
.59083
.38979
.60023
.39832
.60947
.40688
S 17
.58143
.38144
.59099
.38994
.60039
.39846
.60962
.40702
12 18
.58159
.38158
.59115
.39008
.60054
.39861
.60977
.40717
16 19
.58175
.38172
.59131
.39022
.60070
.39875
.60992
.40731
20 20
9.58191
0.38186
9.59147
0.39036
9.60085
0.39889
9.61008
0.40745
24 21
.58207
.38200
.59162
.39050
.60101
.39903
.61023
.40760
..',s' 22
.58223
.38215
.59178
.39064
.60116
.39918
.61038
.40774
32 23
.58239
.38229
.59194
.39079
.60132
.39932
.61053
.40788
36 24
.58255
.38243
.59210
.39093
.60147
.39946
.61069
.40802
40 25
9.58271
0.38257
9.59225
0.39107
9.60163
0.39960
9.61084
0.40817
44 26
.58287
.38271
.59241
.39121
.60178
.39975
.61099
.40831
45 27
.58303
.38285
.59257
.39135
.60194
.39989
.61114
.40845
52 28
.58319
.38299
.59273
.39150
.60209
.40003
.61129
.40860
56 29
.58335
.38314
.59289
.39164
.60225
.40017
.61145
.40874
s '
5* 6m 76°
5h icr 77°
5ft 14™ 78°
5* IS"1 79°
0 30
9.58351
0.38328
9.59304
0.39178
9.60240
0.40032
9.61160 10.40888
4 31
.58367
.38342
.59320
.39192
.60256
.40046
.61175
.40903
5 32
.58383
.38356
.59336
.39206
.60271
.40060
.61190
.40917
12 33
.58399
.38370
.59351
.39221
.60287
.40074
.61205
.40931
16 34
.58415
.38384
.59367
.39235
.60302
.40089
.61221
.40945
20 35
9.58431
0.38398
9.59383
0.39249
9.60318
0.40103
9.61236
0.40960
24 36
.58447
.38413
.59399
.39263
.60333
.40117
.61251
.40974
28 37
.58463
.38427
.59414
.39277
.60348
.40131
.61266
.40988
32 38
.58479
.38441
.59430
.39292
.60364
.40146
.61281
.41003
36 39
.58495
.38455
.59446
.39306
.60379
.40160
.61296
.41017
40 40
9.58511
0.38469
9.59461
0.39320
9.60395
0.40174
9.61312
0.41031
44 41
.58527
.38483
.59477
.39334
.60410
.40188
.61327
.41046
48 42
.58543
.38498
.59493
.39348
.60426
.40203
.61342
.41060
52 43
.58559
.38512
.59508
.39363
.60441
.40217
.61357
.41074
56 44
.58575
.38526
.59524
.39377
.60456
.40231
.61372
.41089
s '
5h jm 76°
5h jjm 77°
5h 15m 78°
gh igm. 79°
6> 45
9.58591
0.38540
9.59540
0.39391
9.60472
0.40245
9.61387
0.41103
4 46
.58607
.38554
.59556
.39405
.60487
.40260
.61402
.41117
5 47
.58623
.38568
.59571
.39420
.60502
.40274
.61417
.41131
12 48
.58639
.38582
.59587
.39434
.60518
.40288
.61433
.41146
16 49
.58655
.38597
.59602
.39448
.60533
.40303
.61448
.41160
20 50
9.58671
0.38611
9.59618
0.39462
9.60549
0.40317
9.61463
0.41174
24 51
.58687
.38625
.59634
.39476
.60564
.40331
.61478
.41189
28 52
.58703
.38639
.59649
.39491
.60579
.40345
.61493
.41203
32 53
.58719
.38653
.59665
.39505
.60595
.40360
.61508
.41217
36 54
.58735
.38667
.59681
.39519
.60610
.40374
.61523
.41232
40 55
9.58750
0.38682
9.59696
0.39533
9.60625
0.40388
9.61538
0.41246
44 56
.58766
.38696
.59712
.39548
.60641
.40402
.61553
.41260
48 57
.58782
.38710
.59728
.39562
.60656
.40417
.61568
.41275
52 58
.58798
.38724
.59743
.39576
.60671
.40431
.61583
.41289
56 59
.58814
.38738
.59759
.39590
.60687
.40445
.61598
.41303
60 60
9.58830
0.38752
9.59774
0.39604
9.60702
0.40460
9.61614
0.41318
270
Table 10. Haversine Table
s '
5h 20™ 80°
5* 24m 81°
5h. 28m 82°
Oh 32'" 83°
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.61614
0.41318
9.62509
0.42178
9.63389
0.43041
9.64253
0.43907
4 1
.61629
.41332
.62524
.42193
.63403
.43056
.64267
.43921
8 2
.61644
.41346
.62538
.42207
.63418
.43070
.64281
.43935
12 3
.61659
.41361
.62553
.42221
.63432
.43085
.64296
.43950
16 4
.61674
.41375
.62568
.42236
.63447
.43099
.64310
.43964
20 5
9.61689
0.41389
9.62583
0.42250
9.63461
0.43113
0.64324
0.43979
24 6
.61704
.41404
.62598
.42264
.63476
.43128
.64339
.43993
28 7
.61719
.41418
.62612
.42279
.63490
.43142
.64353
.44008
32 8
.61734
.41432
.62627
.42293
.63505
.43157
.64367
.44022
36 9
.61749
.41447
.62642
.42308
.63519
.43171
.64381
.44036
40 10
9.61764
0.41461
9.62657
0.42322
9.63534
0.43185
9.64396
0.44051
44 11
.61779
.41475
.62671
.42336
.63548
.43200
.64410
.44065
48 12
.61794
.41490
.62686
.42351
.63563
.43214
.64424
.44080
52 13
.61809
.41504
.62701
.42365
.63577
.43229
.64438
.44094
56 14
.61824
.41518
.62716
.42379
.63592
.43243
.64452
.44109
s '
5h 21m 80°
gh 25m 81°
5h 29m 82°
gh 33m 83°
0 15
9.61839
0.41533
9.62730
0.42394
9.63606
0.43257
9.64467
0.44123
4 16
.61854
.41547
.62745
.42408
.63621
.43272
.64481
.44138
S 17
.61869
.41561
.62760
.42423
.63635
.43286
.64495
.44152
12 18
.61884
.41576
.62774
.42437
.63649
.43301
.64509
.44166
J6 19
.61899
.41590
.62789
.42451
.63664
.43315
.64523
.44181
20 20
9.61914
0.41604
9.62804
0.42466
9.63678
0.43330
9.64538
0.44195
24 21
.61929
.41619
.62819
.42480
.63693
.43344
.64552
.44210
25 22
.61944
.41633
.62833
.42494
.63707
.43358
.64566
.44224
32 23
.61959
.41647
.62848
.42509
.63722
.43373
.64580
.44239
30 24
.61974
.41662
.62863
.42523
.63736
.43387
.64594
.44253
40 25
9.61989
0.41676
9.62877
0.42538
9.63751
0.43402
9.64609
0.44268
44 26
.62003
.41690
.62892
.42552
.63765
.43416
.64623
.44282
4S 27
.62018
.41705
.62907
.42566
.63779
.43430
.64637
.44296
52 28
.62033
.41719
.62921
.42581
.63794
.43445
.64651
.44311
56 29
.62048
.41733
.62936
.42595
.63808
.43459
.64665
.44325
s '
5h 22™ 80°
5* 26m 81°
5h 30m 82°
5h 34m 83°
0 30
9.62063
0.41748
9.62951
0.42610
9.63823
0.43474
9.64679
0.44340
4 31
.62078
.41762
.62965
.42624
.63837
.43488
.64694
.44354
8 32
.62093
.41776
.62980
.42638
.63851
.43503
.64708
.44369
12 33
.62108
.41791
.62995
.42653
.63866
.43517
.64722
.44383
16 34
.62123
.41805
.63009
.42667
.63880
.43531
.64736
.44398
20 35
9.62138
0.41819
9.63024
0.42681
9.63895
0.43546
9.64750
0.44412
24 36
.62153
.41834
.63039
.42696
.63909
.43560
.64764
.44427
28 37
.62168
.41848
.63063
.42710
.63923
.43575
.64778
.44441
32 38
.62182
.41862
.63068
.42725
.63938
.43589
.64793
.44455'
36 39
.62197
.41877
.63082
.42739
.63952
.43603
.64807
.44470
40 40
9.62212
0.41891
9.63097
0.42753
9.63966
0.43618
9.64821
0.44484
44 41
.62227
.41905
.63112
.42768
.63981
.43632
.64835
.44499
48 42
.62242
.41920
.63126
.42782
.63995
.43647
.64849
.44513
52 43
.62257
.41934
.63141
.42797
.64010
.43661
.64863
.44528
56 44
.62272
.41949
.63156
.42811
.64024
.43676
.64877
.44542
s '
5h 23>n 80°
5h 27m 81°
5>* 31m 82°
5* S5m 83°
0 45
9.62287
0.41963
9.63170
0.42825
9.64038
0.43690
9.64891
0.44557
4 46
.62301
.41977
.63185
.42840
.64053
.43704
.64905
.44571
5 47
.62316
.41992
.63199
.42854
.64067
.43719
.64919
.44586
12 48
.62331
.42006
.63214
.42869
.64081
.43733
.64934
.44600
16 49
.62346
.42020
.63228
.42883
.64096
.43748
.64948
.44614
20 50
9.62361
0.42035
9.63243
0.42897
9.64110
0.43762
9.64962
0.44629
24 51
.62376
.42049
.63258
.42912
.64124
.43777
.64976
.44643
•> V KO
.< i OZ
.62390
.42063
.63272
.42926
.64139
.43791
.64990
.44658
32 53
.62405
.42078
.63287
.42941
.64153
.43805
.65004
.44672
36 54
.62420
.42092
.63301
.42955
.64167
.43820
.65018
.44687
40 55
9.62435
0.42106
9.63316
0.42969
9.64181
0.43834
9.65032
0.44701
44 56
.62450
.42121
.63330
.42984
.64196
.43849
.65046
.44716
48 57
.62464
.42135
.63345
.42998
.64210
.43863
.65060
.44730
52 58
.62479
.42150
.63360
.43013
.64224
.43878
.65074
.44745
56 59
.62494
.42164
.63374
.43027
.64239
.43892
.65088
.44759
60 60
9.62509
0.42178
9.63389
0.43041
9.64253
0.43907
9.65102
0.44774
Table 10. Haversine Table
271
s '
5* 36'" 84°
5* 40m 85°
5A 44m 86°
5* 48m 87"
Hav.
No.
Hav.
No.
Hav.
No.
Hav.
No.
0 0
9.65102
0.44774
9.65937
0.45642
9.66757
0.46512
9.67562
0.47383
4 1
.65116
.44788
.65950
.45657
.66770
.46527
.67576
.47398
8 2
.65130
.44803
.65964
.45671
.66784
.46541
.67589
.47412
12 3
.65144
.44817
.65978
.45686
.66797
.46556
.67602
.47427
16 4
.65158
.44831
.65992
.45700
.66811
.46570
.67616
.47441
20 5
9.65172
0.44846
9.66006
0.45715
9.66824
0.46585
9.67629
0.47456
24 6
.65186
.44860
.66019
.45729
.66838
.46599
.67642
.47470
28 7
.65200
.44875
.66033
.45744
.66851
.46614
.67656
.47485
32 8
.65214
.44889
.66047
.45758
.66865
.46628
.67669
.47499
36 9
.65228
.44904
.66061
.45773
.66878
.46643
.67682
.47514
40 10
9.65242
0.44918
9.66074
0.45787
9.66892
0.46657
9.67695
0.47528
44 11
.65256
.44933
.66088
.45802
.66905
.46672
.67709
.47543
48 12
.65270
.44947
.66102
.45816
.66919
.46686
.67722
.47558
52 13
.65284
.44962
.66116
.45831
.66932
.46701
.67735
.47572
56 14
.65298
.44976
.66129
.45845
.66946
.46715
.07748
.47587
s '
5h 37m 84°
5* 41m 85°
5h 45m 86°
,-;'< 4<>m 87°
0 15
9.65312
0.44991
9.66143
0.45860
9.66959
0.46730
9.67762
0.47601
4 16
.65326
.45005
.66157
.45874
.66973
.46744
.67775
.47616
S 17
.65340
.45020
.66170
.45889
.66986
.46759
.67788
.47630
12 18
.65354
.45034
.66184
.45903
.67000
.46773
.67801
.47645
16 19
.65368
.45048
.66198
.45918
.67013
.46788
.67815
.47659
20 20
9.65382
0.45063
9.66212
0.45932
9.67027
0.46802
9.67828
0.47674
24 21
.65396
.45077
.66225
.45947
.67040
.46817
.67841
.47688
25 22
.65410
.45092
.66239
.45961
.67054
.46831
.67854
.47703
32 23
.65424
.45106
.66253
.45976
.67067
.46846
.67868
.47717
30 24
.65438
.45121
.66266
.45990
.67081
.46860
.67881
.47732
40 25
9.65452
0.45135
9.66280
0.46005
9.67094
0.46875
9.67894
0.47746
44 26
.65466
.45150
.66294
.46019
.67108
.46890
.67907
.47761
45 27
.65480
.45164
.66307
.46034
.67121
.46904
.67920
.47775
52 28
.65493
.45179
.66321
.46048
.67134
.46919
.67934
.47790
56 29
.65507
.45193
.66335
.46063
.67148
.46933
.67947
.47805
s '
5* 38m 84°
5* 42™ 85°
5h 4(>'" 86°
')h 50m 87°
0 30
9.65521
0.45208
9.66348
0.46077
9.67161
0.46948
V). 67960
0.47819
4 31
.65535
.45222
.66362
.46092
.67175
.46962
.67973
.47834
8 32
.65549
.45237
.66376
.46106
.67188
.46977
.67986
.47848
J2 33
.65563
.45251
.66389
.46121
.67202
.46991
.68000
.47863
/6 34
.65577
.45266
.66403
.46135
.67215
.47006
.68013
.47877
20 35
9.65591
0.45280
9.66417
0.46150
9.67228
0.47020
9.68026
0.47892
24 36
.65605
.45295
.66430
.46164
.67242
.47035
.68039
.47906
28 37
.65619
.45309
.66444
.46179
.67255
.4704&
.68052
.47921
32 38
.65632
.45324
.66458
.46193
.67269
.47064
.68066
.47935
36 39
.65646
.45338
.66471
.46208
.67282
.47078
.68079
.47950
40 40
9.65660
0.45353
9.66485
0.46222
9.67295
0.47093
9.68092
0.47964
44 41
.65674
.45367
.66499
.46237
.67309
.47107
.68105
.47979
48 42
.65688
.45381
.66512
.46251
.67322
.47122
.68118
.47993
52 43
.65702
.45396
.66526
.46266
.67336
.47136
.68131
.48008
50 44
.65716
.45410
.66539
.46280
.67349
.47151
.68144
.48022
s '
5h 39™ 84°
5* 43m 85°
5h 47m 86"
6* 51m 87°
0 45
9.65729
0.45425
9.66553
0.46295
9.67362
0.47165
9.68158
0.48037
4 46
.65743
.45439
.66567
.46309
.67376
.47180
.68171
.48052
5 47
.65757
.45454
.66580
.46324
.67389
.47194
.68184
.48066
J2 48
.65771
.45468
.66594
.46338
.67402
.47209
.68197
.48081
16 49
.65785
.45483
.66607
.46353
.67416
.47223
.68210
.48095
20 50
9.65799
0.45497
9.66621
0.46367
9.67429
0.47238
9.68223
0.48110
24 51
.65812
.45512
.66635
.46382
.67443
.47252
.68236
.48124
28 52
.65826
.45526
.66648
.46396
-.67456
.47267
.68249
.48139
32 53
.65840
.45541
.66662
.46411
.67469
.47282
.68263
.48153
30 54
.65854
.45555
.66675
.46425
.67483
.47296
.68276
.48168
40 55
9.65868
0.45570
9.66689
0.46440
9.67496
0.47311
9.68289
0.48182
44 56
.65881
.45584
.66702
.46454
.67509
.47325
.68302
.48197
48 57
.65895
.45599
.66716
.46469
.67522
.47340
.68315
.48211
52 58
.65909
.45613
.66730
.46483
.67536
.47354
.68328
.48226
56 59
.65923
.45628
.66743
.46498
.67549
.47369
.68341
.48241
60 60
9.65937
0.45642
9.66757
0.46512
9.67562
0.47383
9.68354
0.48255
272
Table 10. Haversine Table
s '
5h 52m 88°
5* 56™ 89°
Qh Q™ 6* 4m
Hav.
No.
Hav.
No.
Hav.
Hav.
0 0
9.68354
0.48255
9.69132
0.49127
0
9.69897
9.70648
4 1
.68367
.48269
.69145
.49142
4
.69910
.70661
5 2
.68380
.48284
.69158
.49156
8
.69922
.70673
12 3
.68393
.48299
.69171
.49171
12
.69935
.70686
16 4
.68407
.48313
.69184
.49186
16
.69948
.70698
20 5
9.68420
0.48328
9.69197
0.49200
20
9.69960
9.70710
24 6
.68433
.48342
.69209
.49215
24
.69973
.70723
28 1
.68446
.48357
.69222
.49229
28
.69985
.70735
32 8
.68459
.48371
.69235
.49244
32
.69998
.70748
36 9
.68472
.48386
.69248
.49258
36
.70011
.70760
40 10
9.68485
0.48400
9.69261
0.49273
40
9.70023
9.70772
44 11
.68498
.48415
.69274
.49287
44
.70036
.70785
48 12
.68511
.48429
.69286
.49302
48
.70048
.70797
£T5> 1 5
' . J-O
.68524
.48444
.69299
.49316
TO
5
52
.70061
.70809
56 14
.68537
.48459
.69312
.49331
56
.70074
.70822
s '
5h 53™ 88°
5h 57»> 89°
£
9
s
Qh jm Qh gm
0 15
9.68550
0.48473
9.69325
0.49346
i
0
9.70086
9.70834
4 16
.68563
.48488
.69338
.49360
M
4
.70099
.70847
8 17
.68576
.48502
.69350
.49375
6
8
.70111
.70859
/2 18
.68589
.48517
.69363
.49389
fe
12
.70124
.70871
16 19
.68602
.48531
.69376
.49404
1
16
.70136
.70884
20 20
9.68615
0.48546
9.69389
0.49418
20
9.70149
9.70896
24 21
.68628
.48560
.69402
.49433
32
CJ
24
.70161
.70908
28 22
.68641
.48575
.69414
.49447
28
.70174
.7092 
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